A comparison of mathematical features of Turkish and American textbook definitions regarding special quadrilaterals

In this study, I analyzed and compared definitions of special quadrilaterals presented in the US and Turkish secondary school mathematics textbooks. To this end, seven textbooks from each country were examined based on the following features of mathematical definitions: form of presentation and mathematical correctness, minimality, the nature of definitions, and the reference sets and geometric properties used in definitions. The results showed that Turkish textbooks include both procedural and structural definitions of quadrilaterals, while procedural definitions are non-existent in the US mathematics textbooks. Meanwhile, all structural definitions and all procedural definitions excluding one are mathematically correct. Besides, in both country textbooks, there exists more number of minimal and inclusive definitions compared to the non-minimal and exclusive ones. Finally, textbooks from both countries mainly use quadrilaterals as reference sets and sides as geometric properties in definitions of special quadrilaterals. The results bring to light the varied opportunities provided to the students by the US and Turkish textbooks to experience and learn definitions of special quadrilaterals.     

The initial treatment of the concept of function in the selected secondary school mathematics textbooks in the US and China

In order to provide insight into cross-national differences in students’ achievement, this study compares the initial treatment of the concept of function sections of Chinese and US textbooks. The number of lessons, contents, and mathematical problems were analyzed. The results show that the US curricula introduce the concept of function one year earlier than the Chinese curriculum and provide strikingly more problems for students to work on. However, the Chinese curriculum emphasizes developing both concepts and procedures and includes more problems that require explanations, visual representations, and problem solving in worked-out examples that may help students formulate multiple solution methods. This result could indicate that instead of the number of problems and early introduction of the concept, the cognitive demands of textbook problems required for student thinking could be one reason for differences in American and Chinese students’ performances in international comparative studies. Implications of these findings for curriculum developers, teachers, and researchers are discussed.     

A cross-national comparison of reform curricula in Korea and the US in terms of cognitive complexity: the case of fraction addition and subtraction

The overall level of conceptual understanding and mathematical proficiency of students has been a matter of increasing national interest in South Korea. Recently, a new edition of mathematics textbooks aligned with the amendment of the 7th national mathematics curriculum has become available for all elementary grade levels. To characterize the current reform efforts in South Korea, this study examined the quality of the mathematical problems in the current version of the Korean reform textbooks (KM 2) compared with the previous version (KM 1) and one representative US reform curriculum text (EM). Webb’s (Research monograph No. 18: Alignment of science and mathematics standards and assessments in four states. National Institute for Science Education, Madison, 1999) depth of knowledge framework and Son and Senk’s (Educ Stud Math 74(2):117–142, 2010) cognitive expectation feature were employed to examine the kind and level of students’ opportunities to learn along with the type of word problems presented in the three sets of materials. Analysis revealed that the KM 2 provided better opportunities for students to learn fraction addition and subtraction than the KM 1 in terms of the depth and breadth of cognitive complexity. However, there was little difference in addressing and developing the meaning of fraction addition and subtraction through word problems. Moreover, compared with the US reform curriculum materials, the KM 2 provided more problems requiring lower depth of knowledge levels than the US counterpart. Implications of these findings for curriculum developers, textbook and learning materials developers, teachers and future researchers are discussed.     

Preservice Teachers’ Conceptual and Procedural Knowledge of Fraction Operations: A Comparative Study of the United States and Taiwan

This study examined (a) the differences in preservice teachers’ procedural knowledge in four areas of fraction operations in Taiwan and the United States, (b) the differences in preservice teachers’ conceptual knowledge in four areas of fraction operations in Taiwan and the United States, and (c) correlation in preservice teachers’ conceptual knowledge and procedural knowledge of fractions in Taiwan and the United States. Participants were preservice teachers (N = 49) in a teacher education program in the United States and comparable Chinese preservice teachers (N = 47). Results indicated that Chinese preservice teachers performed better in procedural knowledge on fraction operations than American preservice teachers. No significant differences were found for conceptual knowledge on fraction division. Further, the correlation in this study showed that for Chinese and American preservice teachers, the relationship between conceptual and procedural knowledge of fraction operations was weak.     

Students' Retention of Mathematical Knowledge and Skills in Differential Equations

This study investigates students' retention of mathematical knowledge and skills in two differential equations classes. Posttests and delayed posttests after 1 year were administered to students in inquiry‐oriented and traditional classes. The results show that students in the inquiry‐oriented class retained conceptual knowledge, as seen by their performance on modeling problems, and retained equal proficiency in procedural problems, when compared with students in the traditionally taught classes. The results of this study add additional support to the claim that teaching for conceptual understanding can lead to longer retention of mathematical knowledge.     

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.     

Upper Bounds on the Covering Number of Galois-Planes with Small Order

Our paper deals with the covering number of finite projective planes which is related to an unsolved question of P. Erdös. An integer linear programming (ILP) formulation of the covering number of finite projective planes is introduced for projective planes of given orders. The mathematical programming based approach for this problem is new in the area of finite projective planes. Since the ILP problem is NP-hard and may involve up to 360.000 boolean variables for the considered problems, we propose a heuristic based on Simulated Annealing. The computational study gives a new insight into the structure of projective planes and their (minimal) blocking sets. This computational study indicates that the current theoretical results may be improved.     

Reasoning-and-Proving in School Mathematics Textbooks

Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of reasoning-and-proving that pertain to any textbook series.     

Analysis of the role of the calculator in Brazilian textbooks

This paper presents the results of an investigation in which 48 textbooks have been analysed, to observe whether and how primary school textbooks approved by the Brazilian national textbook programme (Programa Nacional do Livro Didático—PNLD) consider the use of calculators. It has been noticed that many textbooks recommend the use of calculators in the classroom and suggest specific activities. In the PNLD teacher’s guide, only in the analysis of some textbooks is there clear information on calculator usage in the classroom. Calculator usage is mainly encouraged in the textbooks of the final years in primary school, mostly with multiplication and division, and distributed in four types of activities: conceptual development, verification of results, results calculation and calculator tool manipulation. Many textbooks have considered the recommendation to use modern technological devices to aid the learning of mathematics, while others still have to assess the value of the calculator as a valid instrument in developing mathematical knowledge.     

The introduction of real numbers in secondary education: an institutional analysis of textbooks

In this paper we analyse the introduction of irrational and real numbers in secondary textbooks, and specifically the propositions on how these should be taught, in a sample of Brazilian textbooks used in state schools and approved by the Ministry of Education. The analyses discussed in this paper follow an institutional perspective (using Chevallard's Anthropological Theory of Didactics). Our results indicate that the notion of irrational number is generally introduced on the basis of the decimal representation of numbers, and that the mathematical need for the construction of the field of real numbers remains unclear in the textbooks. It seems that textbooks used in secondary teaching institutions develop mathematical organisations which focus on the practical block.     

Developing flexible procedural knowledge in undergraduate calculus

Mathematics experts often choose appropriate procedures to produce an efficient or elegant solution to a mathematical task. This flexible procedural knowledge distinguishes novice and expert procedural performances. This article reports on an intervention intended to aid the development of undergraduate calculus students’ flexible use of procedures. Two sections of the same course were randomly assigned to treatment and control conditions. Treatment students completed an assignment on which they resolved derivative-finding problems with alternative methods and compared the two resulting solutions. Control students were assigned a list of functions to differentiate. On the post-intervention test, treatment students were more likely to use a variety of solution methods without prompting than the control. Moreover, the set of treatment section solutions were closer to those of a group of mathematics experts. This study presents evidence that not only is flexible procedural knowledge a key skill in tertiary mathematics, it can be taught.     

Comparing the Development of Fractions in the Fifth‐ and Sixth‐Graders' Textbooks of Singapore,Taiwan, and the USA

This research examined the presentation of fractions in textbooks used by fifth and sixth graders in Singapore, Taiwan, and the United States. The specific textbooks examined were My Pals Are Here! Maths (MPHM) in Singapore; Kung Hsung (KH) in Taiwan; and Mathematics in Context (MiC) in the USA. Results show the problems posed in MiC put more emphasis on real‐life situations than KH textbooks in Taiwan and MPHM in Singapore. Designing materials that provide opportunities to connect mathematics content with applications in real life is consistent with recommendations from professional organizations. The activities in KH and MPHM tended to emphasize procedures, while the activities of MiC focused more on conceptual understanding and less on the development of procedures. An examination of the mathematics textbooks revealed that MPHM introduced and developed fractions the earliest among the three countries investigated and the content taught in MPHM was about one grade earlier than when the same content was experienced by students in KH and MiC.     

A Comparative Analysis of the Addition and Subtraction of Fractions in Textbooks from Three Countries

In this paper, we report on a comparison of the treatment of addition and subtraction of fractions in primary mathematics textbooks used in Cyprus, Ireland, and Taiwan. To this end, we use a framework specifically developed to investigate the learning opportunities afforded by the textbooks, particularly with respect to the presentation of the content and the textbook expectations as manifested in the associated tasks. We found several similarities and differences among the textbooks regarding the topics included and their sequencing, the constructs of fractions, the worked examples, the cognitive demands of the tasks, and the types of responses required of students. The findings emphasized the need to examine textbooks in order to understand differences in instruction and achievement across countries. Indeed, we postulate that within a given country there may exist a recognizable “textbook signature.” We also draw on the results and the challenges inherent in our analysis to provide suggestions and directions for future textbook analysis studies.     

Normalising geometrical figures: dynamic manipulation and construction of meanings for ratio and proportion

Enlarging-shrinking geometrical figures by 13 year-olds is studied during the implementation of proportional geometric tasks in the classroom. Students worked in groups of two using ‘Turtleworlds’, a piece of geometrical construction software which combines symbolic notation, through a programming language, with dynamic manipulation of geometrical objects by dragging on sliders representing variable values. In this paper we study the students’ normalising activity, as they use this kind of dynamic manipulation to modify ‘buggy’ geometrical figures while developing meanings for ratio and proportion. We describe students’ normative actions in terms of four distinct Dynamic Manipulation Schemes (Reconnaissance, Correlation, Testing, Verification). We discuss the potential of dragging for mathematical insight in this particular computational environment, as well as the purposeful nature of the task which sets up possibilities for students to appreciate the utility of proportional relationships.     

Heuristic solutions and confidence intervals for the multicovering problem

The multicovering problem can be expressed as: Minimize CX subject to AX ⩾, b, X ϵ {0, 1}, where A is a zero-one matrix and b is a vector of positive integers. This mathematical model has many applications to scheduling and location problems. Large examples of such problems arise in industrial, commercial and military settings, and their size frequently exceeds the limits of computational tractability. For this important problem, we examine a variety of simple heuristic approaches which can be applied when optimal solutions are not available. The approximate solutions thus generated are used to construct confidence intervals for the unknown value of the optimal solution. A large-scale computational study for randomly generated problmes suggests that these intervals are both very narrow and very likely to contain the optimal solution value. A study of 10 very large real-world problems further supports the success of our methodology and the quality of the approximate solutions found.     

The <Emphasis Type="Italic">p</Emphasis>-Median and its Linear Programming Relaxation: An Approach to Large Problems

A revised mathematical programme is presented which reduces the number of rows and columns in the Linear Programming formulation of the p-Median Problem. The revised formulation reduces both the computer space required and computational time allowing the solution of large problems while still guaranteeing optimality. Computational experience with the new formulation is presented.     

Parallelization Strategies for Rollout Algorithms

Rollout algorithms are innovative methods, recently proposed by Bertsekas et al. [3], for solving NP-hard combinatorial optimization problems. The main advantage of these approaches is related to their capability of magnifying the effectiveness of any given heuristic algorithm. However, one of the main limitations of rollout algorithms in solving large-scale problems is represented by their computational complexity. Innovative versions of rollout algorithms, aimed at reducing the computational complexity in sequential environments, have been proposed in our previous work [9]. In this paper, we show that a further reduction can be accomplished by using parallel technologies. Indeed, rollout algorithms have very appealing characteristics that make them suitable for efficient and effective implementations in parallel environments, thus extending their range of relevant practical applications.We propose two strategies for parallelizing rollout algorithms and we analyze their performance by considering a shared-memory paradigm. The computational experiments have been carried out on a SGI Origin 2000 with 8 processors, by considering two classical combinatorial optimization problems. The numerical results show that a good reduction of the execution time can be obtained by exploiting parallel computing systems.     

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.     

A Richardson-type iterative approach for identification of delamination boundaries

A direct problem of applied mathematical modelling is to determine the response of a system given the governing partial differential equations, the geometry of interest, the complete boundary and initial conditions, and material properties. When one or more of the conditions for the solution of the direct problem are unknown, an inverse problem can be formulated. One of the methods frequently used for the solution of inverse problems involves finding the values of the unknowns in a mathematical formulation such that the behavior calculated with the model matches the measured response to a degree evaluated in terms of the classical L ₂ norm. Considered in this sense, the inverse problem is equivalent to an ill-posed optimization problem for the estimation of parameters whose solution in the majority of cases is a real mathematical challenge. In this contribution, we report a novel approach that avoids the mathematical difficulties inspired by the ill-posed character of the model. Our method is devoted to the computation of inverse problems furnished by second-order elliptical systems of partial differential equations and falls in the same conceptual line with the method initiated by Kozlov et al. and further extended and algorithmized by Weikl et al. We construct and employ a weak version of the algorithm found by Weikl et al. Proofs for the convergence and regularity of this version are given for the case of a single layer. The computational realization of the algorithm (called briefly AICRA) is applied and numerical results are obtained. Comparison with experiments demonstrates a good significance and representativeness. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 209–230, 2006.     

Solving large quadratic assignment problems on computational grids

The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computational platforms. In this article we describe a novel approach to solve QAPs using a state-of-the-art branch-and-bound algorithm running on a federation of geographically distributed resources known as a computational grid. Solution of QAPs of unprecedented complexity, including the nug30, kra30b, and tho30 instances, is reported. Received: September 29, 2000 / Accepted: June 5, 2001?Published online October 2, 2001