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.     

Influence of a Mathematics Teachers' Circle on Elementary Teachers' Use of Problem Solving

Math teachers' circles are a form of professional development that is recommended by the Conference Board of the Mathematical Sciences in their publication Mathematical Education of Teachers II (2012). However, little research has been published on how effective math teachers' circles are in advancing the mathematical knowledge of teachers and influencing their actual teaching practice. This study reports how 25 elementary teachers from schools with minority and economically disadvantaged populations responded to a math circle experience. A pretest and posttest showed a significant increase in problem‐solving ability. Through analysis of journals, all but 4 of the 25 teachers indicated that participation in the math circle increased their understanding of problem solving and advanced their use of problem solving in the classroom.     

Korean and American Elementary School Teachers' Beliefs about Mathematics Problem Solving

Elementary school teachers in South Korea and the United States completed a beliefs and practices questionnaire pertaining to mathematical problem-solving instruction. Although both groups of teachers shared a general approach to teaching with a focus on problem-solving strategies, many differences were apparent. Korean teachers rated themselves and their students higher in problem-solving ability than American teachers. Korean teachers perceived their mathematics textbook as a more valuable source for problem-solving instruction and word problems. Korean teachers more strongly agreed that students should know the key-word approach for solving problems. American teachers reported more frequent use of calculators, manipulatives, and small group instruction. The results indicate that American teachers may more often use instructional techniques that are aligned with current recommendations for mathematics instruction.     

Two Probabilistic Models for Solving an Oxygen-bottle Inventory Problem

In this paper, an inventory oxygen-bottles problem is formulated as (i) an (M/M/C):(GD/N/∞) queueing model with different service rates; (ii) a probabilistic goal programming (PGP) model. By using the first model, the optimum required rate for oxygen bottles to be stored is determined. By using the second model, the decision-maker can determine his policy, which minimizes the total cost. Finally, a case study of the storage of oxygen bottles for the Egyptian Air Force (1980-1984) is presented.     

The Impact of Enacted Mathematics Curriculum Models on Prospective Elementary Teachers' Course Perceptions and Beliefs

This paper communicates the impact of prospective teachers' learning of mathematics using novel curriculum materials in an innovative classroom setting. Two sections of a mathematics content course for prospective elementary teachers used different text materials and instructional approaches. The primary mathematical authorities were the instructor and text in the textbook section and the prospective teachers in the curriculum materials section. After one semester, teachers in the curriculum materials section (n= 34) placed significantly more importance on classroom group work and discussions, less on instructor lecture and explanation, and less on textbooks having practice problems, examples, and explanations. They valued student exploration over practice. In the textbook section (n= 19), there was little change in the teachers' beliefs, in which practice was valued over exploration. These results highlight the positive impact of experiences with innovative curriculum materials on prospective elementary teachers' beliefs about mathematics instruction.     

Making Connections: Elementary Teachers' Construction of Division Word Problems and Representations

If teachers make few connections among multiple representations of division, supporting students in using representations to develop operation sense demanded by national standards will not occur. Studies have investigated how prospective and practicing teachers use representations to develop knowledge of fraction division. However, few studies examined primary (K‐3) teachers' learning of contextual division problems, making connections among representations of division, and resolving the ambiguity of representing quotients with remainders. A written post‐course assessment provided evidence that most teachers created partitive division word problems, used a set model without splitting the remainder, and wrote equations with limited success. Post‐course written reflections demonstrated that many teachers developed pedagogical knowledge for helping students make connections among multiple representations, and mathematical knowledge of unit fractions. These findings suggest two areas that have implications for mathematics teacher educators who design professional development courses to facilitate teachers' learning of mathematical content and pedagogical knowledge of division and fraction relationships.     

Area and Perimeter: Preservice Teachers' Confusion

Measurement topics, including perimeter and area, are basic to the competency domain of elementary teachers. This study examines solution strategies, used by elementary preservice teachers, to find the perimeter and area of a shaded geometric figure. Many of these preservice teachers incorrectly found the perimeter of the figure by the same method they used to find the area. Examination of incorrect strategies by preservice teachers provides an opportunity to investigate inadequacies in the mathematical backgrounds of these students as well as to make curriculum changes in teacher education programs.     

Solving the Problem of Rerostering Nurse Schedules with Hard Constraints: New Multicommodity Flow Models

The problem of rerostering nurse schedules arises in hospitals when at least one nurse informs that she will be unable to perform the shifts assigned to her on one or more future work days. As a result, the current roster must be rebuilt in accordance with labour contract rules and institutional requirements. All such restraints are regarded as hard constraints. However, major alterations in the previously assigned nurse schedules must be avoided. This paper is based on a case study of a public hospital in Portugal. It presents two new integer multicommodity flow formulations for the rerostering problem, besides a computational experiment performed using real data. The first model is based on a directed multilevel acyclic network. The aggregation of nodes in this network led to the second model. The results obtained show that the second integer multicommodity flow formulation outperforms the first, both in terms of solution quality, as well as in computational time.