Rhetoric,Reality, and Possibilities: Interdisciplinary Teaching and Secondary Mathematics

The National Council of Teachers of Mathematics has proposed a broad core mathematics curriculum for all high school students. One emphasis in that core is on “mathematical connections” both among mathematical topics and between mathematics and other disciplines of study. It is suggested that mathematics should become a more integrated part of all students' high school education. In this article, working definitions for the terms curriculum, interdisciplinary, and integrated and a model of three categories of curriculum design based on the work of Harold Alberty are developed. This article then examines how a “connected” mathematics core curriculum might be situated within the different categories of curriculum organization. Examples from research on interdisciplinary education in high schools are presented. Issues arising from this study suggest the need for a greater emphasis on building and using models of curriculum integration both to frame and to give impetus to the work being done by teachers and administrators.     

Integration of Science and Mathematics: A Theoretical Model

Interest in interdisciplinary, integrated curriculum development continues to increase. However, teachers, who have been given primary responsibility for developing these materials, are often working with little guidance. At present there exists no clear definition of the meaning of integration of mathematics and science. A continuum model of integration is proposed as a useful tool for curriculum developers as they create new integrated mathematics and science curricula or adapt commercially prepared materials. On the continuum, activities range from mathematics or science involving no integration to those activities including balanced mathematics and science concepts. Several examples are given to illustrate the utility of the continuum model for analyzing integrated curricula. The continuum model is intended to be used by curriculum developers to clarify the relationship between the mathematics and science activities and concepts and to guide the modification of lessons.     

Problem Solving: Teachers' Perceptions,Content Area Models,and Interdisciplinary Connections

Features of common problem-solving models in mathematics and science, as well as those found in business and industry today, are discussed. Commonalties in the models are used to advance a case for interdisciplinary or integrated instruction in mathematics, science and technology. The Integrated Mathematics, Science and Technology (IMaST) program's problem-solving model is presented as an example of a curriculum project that draws upon the commonalties in the problem-solving models as a basis for a seventh grade integrated curriculum.     

Elementary Science and Language Arts: Should We Blur the Boundaries?

Elementary school goals for instruction focus on developing literate readers and writers. It has been recommended that language arts strategies can help elementary teachers more effectively teach science. The terms “integrated, interdisciplinary, and thematic instruction” are defined and examples are given for using each in an elementary classroom. Definitions are provided comparing language arts and scientific literacy. Use of thematic instruction with an interdisciplinary focus is recommended to help meet both language arts and science goals and objectives as they relate to the National Science Education Standards and the National English Language Arts Standards. Recommendations are made for helping teachers effectively use language arts strategies to help develop science literacy, and science to provide purpose for reading and writing activities within thematic, interdisciplinary instruction often found in elementary schools.     

Developing Interdisciplinary Units: Strategies and Examples

The authors use a common theme of SHARKS to illustrate the process of developing an interdisciplinary unit for middle school instruction. A model is presented for teams to use in developing integrated curriculum. Focusing on a central theme or topic, a web of disciplines and key concepts is developed. As activities evolve, a concept map is created to illustrate the relationships and integration of ideas and activities.     

Developing Interdisciplinary Units: A Strategy Based on Problem Solving

While the benefits of the interdisciplinary unit are well documented, it presents a complex challenge to teachers in the natural and social sciences, mathematics, and humanities. Teachers must become active curriculum designers who shape and edit the curriculum according to students' needs. This paper describes knowledge for teachers as curriculum designers and a framework for interdisciplinary unit development. The framework addresses a metacurricular process (problem solving) that will be the unit centerpiece, the development of this central process related to the learner, and the tasks that teach explicit learning and thinking skills attached to the central process. An example of the framework in action is also described. As the faculty and curriculum coordinators for an innovative summer academy for minority students in northern Arizona have used this framework, they have evolved from a group that created a good idea to interest students with parallel subject development in separate classrooms to humanities/mathematics/science teams united in one team/classroom, in which content is integrated through the actions of the problem solving process.     

Revealing hidden mathematics curriculum to pre-teachers using technology: the case of partitions

Motivated by work done with pre-teachers of mathematics in a problem-solving course, this paper shows how computing technologies, including a spreadsheet and Maple, facilitate an informal journey into a hidden aspect of the formal content of the pre-college curriculum dealing with the arithmetic of partitions. By using three problems from different grade levels within a state curriculum as an example, the paper suggests that a deeper perspective on seemingly disconnected problem-solving contexts may serve as a powerful didactical tool in helping teachers to appreciate mathematics and its pedagogy as an integrated whole. The connection of the hidden aspect of the curriculum to the concept of mathematical play is also explored.     

Configurational-Force-Based Adaptive FE Solver for a Phase Field Model of Fracture

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by diffusive crack modeling, based on the introduction of a crack phase field as outlined in [1, 2]. Following these formulations, we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles and, as an extension to [1, 2], consider their numerical implementations by an efficient h-adaptive finite element method. A key problem of the phase field formulation is the mesh density, which is required for the resolution of the diffusive crack patterns. To this end, we embed the computational framework into an adaptive mesh refinement strategy that resolves the fracture process zones. We construct a configurational-force-based framework for h-adaptive finite element discretizations of the gradient-type diffusive fracture model. We develop a staggered computational scheme for the solution of the coupled balances in physical and material space. The balance in the material space is then used to set up indicators for the quality of the finite element mesh and accounts for a subsequent h-type mesh refinement. The capability of the proposed method is demonstrated by means of a numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Why are there only three quantum noises?

In quantum stochastic calculus on the symmetric Fock space over L ²(ℝ₊), adapted processes of operators are integrated with respect to creation, annihilation and number processes. The main property which allows this integration is that the increments of integrators between s and t act only on Fock space over L ²([s, t]). In this article, we prove that there are no other process of closable operators on coherent vectors with this property. Thus the only possible integrators in quantum stochastic calculus are the creation, annihilation and number processes.     

Configurational-force-based structural updates and adaptive remeshing strategies

The application of configurational forces in h -adaptive strategies for fracture mechanics and inelasticity is investigated. Starting from a global Clausius-Planck inequality, dual equilibrium conditions are derived by means of a Coleman-type exploitation method. The remaining reduced dissipation inequality is used for the derivation of evolution equations for the internal variables. In fracture mechanics, crack loading conditions as well as a normality rule for the crack propagation are obtained. In the discrete setting, the crack propagation is governed by a configurational-force-driven update of the geometry model. The material balance equation is used to set up a h -adaptive refinement indicator. A relative global criterion is defined used for the decision on mesh refinement. In addition, a criterion on the element level is evaluated controlling the local refinement procedure. The capability of the proposed procedures is demonstrated by means of numerical examples. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Teachers' Conceptions of Integrated Mathematics Curricula

In this qualitative research study, we sought to understand teachers' conceptions of integrated mathematics. The participants were teachers in the first year of implementation of a state‐mandated, high school integrated mathematics curriculum. The primary data sources for this study included focus group and individual interviews. Through our analysis, we found that the teachers had varied conceptions of what the term integrated meant in reference to mathematics curricula. These varied conceptions led to the development of the Conceptions of Integrated Mathematics Curricula Framework describing the different conceptions of integrated mathematics held by the teachers. The four conceptions—integration by strands, integration by topics, interdisciplinary integration, and contextual integration—refer to the different ideas teachers connect as well as the time frame over which these connections are emphasized. The results indicate that even when teachers use the same integrated mathematics curriculum, they may have varying conceptions of which ideas they are supposed to connect and how these connections can be emphasized. These varied conceptions of integration among teachers may lead students to experience the same adopted curriculum in very different ways.     

Mathematics curriculum: a vehicle for school improvement

Different forms of curriculum determine what is taught and learned in US classrooms and have been used to stimulate school improvement and to hold school systems accountable for progress. For example, the intended curriculum reflected in standards or learning expectations increasingly influences how instructional time is spent in classrooms. Curriculum materials such as textbooks, instructional units, and computer software constitute the textbook curriculum, which continues to play a dominant role in teachers’ instructional decisions. These decisions influence the actual implemented curriculum in classrooms. Various curriculum policies, including mandated end-of-course assessments (the assessed curriculum) and requirements for all students to complete particular courses (e.g., year-long courses in algebra, geometry, and advanced algebra or equivalent integrated mathematics courses) are also being implemented in increasing numbers of states. The wide variation across states in their intended curriculum documents and requirements has led to a historic and precedent-setting effort by the Council of Chief State School Officers and the National Governors Association Council for Best Practices to assist states in the development and adoption of common College and Career Readiness Standards for Mathematics. Also under development by this coalition is a set of common core state mathematics standards for grades K-12. These sets of standards, together with advances in information technologies, may have a significant influence on the textbook curriculum, the implemented curriculum, and the assessed curriculum in US classrooms in the near future.     

Design and Implementation of a Framework for Defining Integrated Mathematics and Science Education

Integrated mathematics and science teaching and learning is a widely advocated yet largely unexplored phenomenon. This study involves an examination of middle school integrated mathematics and science education from two perspectives—theory and practice. The theoretical component of this research addresses the ill-defined nature of the phrase integrated mathematics and science education. A conceptual framework in the form of a Mathematics/Science Continuum is presented to lend clarity and precision to this phrase. The theoretical framework is then used to guide analysis of tasks students are engaged in during instructional practice in middle school classrooms, where the goal of instruction is full integration of mathematics and science. Barriers to integrating mathematics and science in the school curriculum are also presented.     

Revisiting mathematical problem solving and posing in the digital era: toward pedagogically sound uses of modern technology

The availability of sophisticated computer programs such as Wolfram Alpha has made many problems found in the secondary mathematics curriculum somewhat obsolete for they can be easily solved by the software. Against this background, an interplay between the power of a modern tool of technology and educational constraints it presents is discussed. Using topics from algebra (equations) and elementary number theory (summation of powers of integers), the paper suggests ways of developing problems that are both technology-immune and technology-enabled in the sense that whereas software can facilitate problem solving, its direct application is not sufficient for finding an answer. Stemming from the author's work with secondary mathematics teacher candidates, this paper highlights the appropriate use of technology as support system for multiple ways of knowing and knowledge construction in the modern classroom.     

Phase coexistence in Ising, Potts and percolation models

We study phase coexistence (separation) phenomena in Ising, Potts and random cluster models in dimensions d3 below the critical temperature. The simultaneous occurrence of several phases is typical for systems with appropriately arranged (mixed) boundary conditions or for systems satisfying certain physically natural constraints (canonical ensembles). The various phases emerging in these models define a partition, called the empirical phase partition, of the space. Our main results are large deviations principles for (the shape of) the empirical phase partition. More specifically, we establish a general large deviation principle for the partition induced by large (macroscopic) clusters in the Fortuin–Kasteleyn model and transfer it to the Ising–Potts model where we obtain a large deviation principle for the empirical phase partition induced by the various phases. The rate function turns out to be the total surface free energy (associated with the surface tension of the model and with boundary conditions) which can be naturally assigned to each reasonable partition. These LDP-s imply a weak law of large numbers: asymptotically, the law of the phase partition is determined by an appropriate variational problem. More precisely, the empirical phase partition will be close to some partition which is compatible with the constraints imposed on the system and which minimizes the total surface free energy. A general compactness argument guarantees the existence of at least one such minimizing partition. Our results are valid for temperatures T below a limit of slab-thresholds conjectured to agree with the critical point T_c. Moreover, T should be such that there exists only one translation invariant infinite volume state in the corresponding Fortuin–Kasteleyn model; a property which can fail for at most countably many values and which is conjectured to be true for every T≠T_c.     

Hamiltonian reduction of free particle motion on the group SL(2, ℝ)

We analyze the structure of the reduced phase space that arises in the Hamiltonian reduction of the phase space of free particle motion over the groupSL(2, ℝ). The reduction considered is based on introducing constraints that are analogous to those used in the reduction of the Wess-Zumino-Novikov-Witten model to Toda systems. It is shown that the reduced phase space is diffeomorphic either to a union of two two-dimensional planes or to a cylinder S¹×ℝ. We construct canonical coordinates for both cases and show that in the first case, the reduced phase space is symplectomorphic to the union of two cotangent bundles T*(ℝ) endowed with a canonical symplectic structure, while in the second case, it is symplectomorphic to the cotangent bundle T* (S¹), which is also endowed with a canonical symplectic structure. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 149–161, January, 1997.     

Supports of Locally Linearly Independent M-Refinable Functions,Attractors of Iterated Function Systems and Tilings

This paper is devoted to a study of supports of locally linearly independent M-refinable functions by means of attractors of iterated function systems, where M is an integer greater than (or equal to) 2. For this purpose, the local linear independence of shifts of M-refinable functions is required. So we give a complete characterization for this local linear independence property by finite matrix products, strictly in terms of the mask. We do this in a more general setting, the vector refinement equations. A connection between self-affine tilings and L ₂ solutions of refinement equations without satisfying the basic sum rule is pointed out, which leads to many further problems. Several examples are provided to illustrate the general theory.     

Student Reactions to Standards-Based Mathematics Curricula: The Interplay Between Curriculum,Teachers, and Students

As standards-based mathematics curricula are used to guide learning, it is important to capture not just data on achievement but data on the way in which students respond to and interact in a standards-based instructional setting. In this study, sixth and seventh graders reacted through letters to using one of two standards-based curriculum projects (Connected Mathematics Project or Six Through Eight Mathematics). Letters were analyzed by class, by teacher, and by curriculum project. Findings suggest that across classrooms students were positive toward applications, hands-on activities, and working collaboratively. The level of students' enthusiasm for the new curricula varied much from class to class, further documenting the critical role teachers play in influencing students' perceptions of their mathematics learning experiences. The results illustrate that, while these curricula contain rich materials and hold much promise, especially in terms of their activities and applications, their success with students is dependent on the teacher.     

Economic design of integrated model of control chart and maintenance management

There are two key tools for the control of production processes — Statistical Process Control (SPC) and Maintenance Management (MM), which are traditionally separated (both in science and in business practice), even though their goals overlap a great deal. Their common goal is to achieve optimal product quality, little downtime and cost reduction by controlling variances in the process. Since single or separated parallel applications may not be fully effective, this paper discusses the integration of statistical process control and maintenance, and provides an integrated model of Control Chart (CC) and MM. A mathematical model is given to analyze the cost of the integrated model and the grid-search approach is used to find the optimal values of policy variables (n,h,L,k) that minimize hourly cost. Finally, a numerical experiment is conducted to investigate the effects of cost parameters on the solution of the design.