Investigating young students’ multiplicative thinking: The 12 little ducks problem

Children’s multiplicative thinking as the visualization of equal group structures and the enumeration the composite units was the subject of this study. The results were obtained from a small sample of Australian children (n = 18) in their first year of school (mean age 5 years 6 months) who participated in a lesson taught by their classroom teacher. The 12 Little Ducks problem stimulated children to visualize and to draw different ways of making equal groups. Fifteen children (83 %) could identify and create equal groups; eight of these children (44 %) could also quantify the number of groups they formed. These findings show that some young children understand early multiplicative ideas and can visualize equal group situations and communicate about these through their drawings and talk. The study emphasises the value of encouraging mathematical visualization from an early age; using open thought-provoking problems to reveal children’s thinking; and promoting drawing as a form of mathematical communication.     

Between Counting and Multiplication: Low-Attaining Students’ Spatial Structuring,Enumeration and Errors in Concretely-Presented 3D Array Tasks

The move from additive to multiplicative thinking requires significant change in children’s comprehension and manipulation of numerical relationships, involves various conceptual components, and can be a slow, multistage process for some. Unit arrays are a key visuospatial representation for supporting learning, but most research focuses on 2D (rectangular) arrays, and when focusing on 3D (cuboid) arrays still frequently uses 2D representations. This article documents low-attaining children’s partially developed multiplicative thinking as they work on concretely presented 3D array tasks; it also presents a framework for microanalysis of learners’ early multiplicative thinking in array tasks. Data derives from a small but cognitively diverse set of participants, all arithmetically low-attaining and relying heavily on counting: this enabled detailed analysis of small but significant differences in their arithmetical engagement with arrays. The analytical framework combines and builds on previous structural and enumerative categorizations, and may be used with a variety of array representations.     

Spontaneous Focusing on Quantitative Relations: Towards a Characterization

In contrast to previous studies on Spontaneous Focusing on Quantitative Relations (SFOR), the present study investigated not only the extent to which children focus on (multiplicative) quantitative relations, but also the nature of children’s quantitative focus (i.e., the types of quantitative relations that children focus on). Therefore, we offered three different SFOR tasks – a multiplicative, additive, or open SFOR task – to 315 second, fourth, and sixth graders. Results revealed, first, that most children spontaneously focused on quantitative relations. Some focused on multiplicative relations, and others on additive relations. Second, SFOR, and especially multiplicative SFOR, increased with grade, while the development of additive SFOR differed between tasks. Third, the open SFOR task seemed best suited to capture SFOR, since it evoked the largest number of each type of relational answers ? while still showing substantial interindividual differences in SFOR. These results indicate that a broader conceptualization and operationalization of SFOR than the unilateral multiplicative one are warranted.     

Notions of equivalence through ratios: Students with and without learning disabilities

Students with learning disabilities (LD) specific to mathematics historically underperform in foundational content such as rational number equivalence. This study examined the strategy usage and multiplicative thinking of three third grade children (i.e., Bill, a child identified as having a learning disability specific to mathematics, Carl, a child labeled as low achieving in mathematics, and Albert, a child labeled as typically achieving) before, during, and after participating in tutoring sessions consisting of student-centered pedagogy and equivalence tasks presented through an underutilized interpretation of rational number: namely, the ratio interpretation. Constant comparison analysis of the children's work during the tutoring sessions as well as responses to tasks during two clinical interviews seemed to indicate that all three children increased their use of viable strategies, with notable differences in the sophistication of the strategies as well as the level of multiplicative thinking utilized before and after the ratio-based tutoring sessions. Yet, Bill's continued use of rudimentary strategies reflects a need for continued research to investigate why the use of such strategies persists and how supporting the development of more sophisticated strategies (especially among children with LD) can be achieved.     

Third- to fourth-grade students’ conceptions of multiplication and area measurement

In this study, 34 children were evaluated in order to elucidate their multiplicative thinking and interpretation of the area formula of a rectangle, and to determine what roles these factors play in solving area measurement problems. One-on-one interviews and problem-solving tasks were employed to explore the problem-solving skills of the children regarding these two concepts. This study also explored how the associations changed throughout two consecutive phases, from the third to the fourth grades. The results indicated that in the third grade, multiplicative thinking was associated with the solving of area measurement problems. Third-grade children who understood the meaning of the multiplication symbol “p × q” in models (e.g., the set model and arrays) outperformed children who understood only partial multiplicative concepts or additive thinking; however, the association between multiplicative thinking and solving area measurement problems was not significant in the fourth grade. In contrast, children’s ability to interpret the area formula of a rectangle was associated with their performance at solving area measurement problems throughout the third and fourth grades. The way of interpreting the area formula was associated with the extent to which the children understood multiplication, area measurement, and the spatial concepts embedded in rectangular figures. The instructional implications of the study are discussed in terms of developing child abilities to solve area measurement problems by connecting multiplication and area measurement.     

Anisotropy-based bounded real lemma for discrete-time systems with multiplicative noise

A model of a discrete-time system with multiplicative noise is considered. For this model, a condition is derived under which the anisotropic norm of the system is bounded by the anisotropic norm of an auxiliary linear discrete-time stationary system with parametric uncertainty. Conditions for the anisotropic norm of the system with multiplicative noise to be bounded by a given positive number are obtained in terms of solutions of linear matrix inequalities and a single equation.     

Global convergence of modified multiplicative updates for nonnegative matrix factorization

Nonnegative matrix factorization (NMF) is the problem of approximating a given nonnegative matrix by the product of two nonnegative matrices. The multiplicative updates proposed by Lee and Seung are widely used as efficient computational methods for NMF. However, the global convergence of these updates is not formally guaranteed because they are not defined for all pairs of nonnegative matrices. In this paper, we consider slightly modified versions of the original multiplicative updates and study their global convergence properties. The only difference between the modified updates and the original ones is that the former do not allow variables to take values less than a user-specified positive constant. Using Zangwill’s global convergence theorem, we prove that any sequence of solutions generated by either of those modified updates has at least one convergent subsequence and the limit of any convergent subsequence is a stationary point of the corresponding optimization problem. Furthermore, we propose algorithms based on the modified updates that always stop within a finite number of iterations.     

Student Strategies Suggesting Emergence of Mental Structures Supporting Logical and Abstract Thinking: Multiplicative Reasoning

For many students, developing mathematical reasoning can prove to be challenging. Such difficulty may be explained by a deficit in the core understanding of many arithmetical concepts taught in early school years. Multiplicative reasoning is one such concept that produces an essential foundation upon which higher‐level mathematical thinking skills are built. The purpose of this study is to recognize indicators of multiplicative reasoning among fourth‐grade students. Through cross‐case analysis, the researcher used a test instrument to observe patterns of multiplicative reasoning at varying levels in a sample of 14 math students from a low socioeconomic school. Results indicate that the participants fell into three categories: premultiplicative, emergent, and multiplier. Consequently, 12 new sublevels were developed that further describe the multiplicative thinking of these fourth graders within the categories mentioned. Rather than being provided the standard mathematical algorithms, students should be encouraged to personally develop their own unique explanations, formulas, and understanding of general number system mechanics. When instructors are aware of their students' distinctive methods of determining multiplicative reasoning strategies and multiplying schemes, they are more apt to provide the most appropriate learning environment for their students.     

Complex dynamic behavior during transition in a solid combustion model

Through examples in a free‐boundary model of solid combustion, this study concerns nonlinear transition behavior of small disturbances of front propagation and temperature as they evolve in time. This includes complex dynamics of period doubling, and quadrupling, and it eventually leads to chaotic oscillations. Within this complex dynamic domain we also observe a period six‐folding. Both asymptotic and numerical solutions are studied.We show that for special parameters our asymptotic method with some dominant modes captures the formation of coherent structures. Finally,we discuss possible methods to improve our prediction of the solutions in the chaotic case. © 2009 Wiley Periodicals, Inc. Complexity, 2009     

A Parameterized Class of Complex Nonsymmetric Algebraic Riccati Equations

In this paper, by introducing a definition of parameterized comparison matrix of a given complex square matrix, the solvability of a parameterized class of complex nonsymmetric algebraic Riccati equations (NAREs) is discussed. The existence and uniqueness of the extremal solutions of the NAREs is proved. Some classical numerical methods can be applied to compute the extremal solutions of the NAREs, mainly including the Schur method, the basic fixed-point iterative methods, Newton's method and the doubling algorithms. Furthermore, the linear convergence of the basic fixed-point iterative methods and the quadratic convergence of Newton's method and the doubling algorithms are also shown. Moreover, some concrete parameter selection strategies in complex number field for the doubling algorithms are also given. Numerical experiments demonstrate that our numerical methods are effective.     

Definition and properties of characteristic frequencies of a linear equation

The definition of the characteristic frequencies of zeroes and changes of sign for solutions is given. It is equal to the upper medium (with respect to the time half-axis) of their number on the half-interval of length π. We also define the main frequencies for a linear homogeneous equation of order n. These main frequencies for an equation with constant coefficients coincide with the absolute values of the imaginary parts of the roots of the corresponding characteristic polynomial. It is proved that for the second-order equation the main frequencies are the same for all solutions and that they are stable with respect to uniformly small and infinitely small perturbations of the coefficients. For the third-order equation they can be different, and for any of the main frequencies an example of nonstability is given. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 249–294, 2005.     

模糊数互补判断矩阵的乘性一致性检验及改进方法

针对模糊数互补判断矩阵的乘性一致性检验及改进问题进行研究。在文献[11]引入模糊数的Q-算子和模糊数互补判断矩阵的Q-算子矩阵概念的基础上,通过构造具有乘性一致性的特征矩阵及偏差矩阵,建立了衡量乘性一致性程度的指标值并用设定阈值的方法给出了满意乘性一致性的概念,对于不满足满意乘性一致性的情况提出了改进方法。最后通过一个算例说明了此方法的可行性。     

Inverse eigenvalue problems

The classical inverse additive and multiplicative inverse eigenvalue problems for matrices are studied. Using general results on the solvability of polynomial systems it is shown that in the complex case these problems are always solvable by a finite number of solutions. In case of real symmetric matrices the inverse problems are reformulated to have a real solution. An algorithm is given to obtain this solution.     

The dynamical behaviors of a Ivlev-type two-prey two-predator system with impulsive effect

In this paper, considering the strategy of integrated Pest Management (IPM), a class of two-prey two-predator system with the Ivlev-type functional response and impulsive effect at different fixed time is established. By using impulsive comparison theorem, Floquent theory and small amplitude perturbation skill, the sufficient conditions for the system to be extinct of prey and permanence are proved. Moreover, we give two sufficient conditions for the extinction of one of two prey and remaining three species are permanent. Numerical simulation shows that there exist complex dynamics for system, such as symmetry-breaking pitchfork bifurcation, periodic doubling bifurcation, chaos, periodic halving cascade. Lastly, a brief discussion is given.     

Topological Entropy and Secondary Folding

A convenient measure of a map or flow’s chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with ‘secondary folding’: material lines undergo folding which is not homologically forced. We examine this phenomenon both for physical rod stirring devices and toral linked twist maps, and show rigorously that for the latter secondary folds occur.     

On the number of halving planes

LetS ³ be ann-set in general position. A plane containing three of the points is called a halving plane if it dissectsS into two parts of equal cardinality. It is proved that the number of halving planes is at mostO(n ^2.998).As a main tool, for every setY ofn points in the plane a setN of sizeO(n ⁴) is constructed such that the points ofN are distributed almost evenly in the triangles determined byY.Research supported partly by the Hungarian National Foundation for Scientific Research grant No. 1812     

Generalized Liouville property for Schrödinger operator on Riemannian manifolds

In this paper, we prove that the dimension of the space of positive (bounded, respectively) -harmonic functions on a complete Riemannian manifold with -regular ends is equal to the number of ends (-nonparabolic ends, respectively). This result is a solution of an open problem of Grigor'yan related to the Liouville property for the Schr?dinger operator . We also prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincaré inequality and the finite covering condition on each end, then the dimension of the space of positive (bounded, respectively,) solutions for the Schr?dinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends (-nonparabolic ends, respectively). This is a partial answer of the question, suggested by Li, related to the regularity of ends of a complete Riemannian manifold. Especially, our results directly generalize various earlier results of Yau, of Li and Tam, of Grigor'yan, and of present authors, but with different techniques that the peculiarity of the Schr?dinger operator demands. Received: 4 April 2000; in final form: 19 September 2000 / Published online: 25 June 2001     

“Approximate” multiplicative relationships between quantitative unknowns

Three 18-session design experiments were conducted, each with 6–9 7th and 8th grade students, to investigate relationships between students’ rational number knowledge and algebraic reasoning. Students were to represent in drawings and equations two multiplicatively related unknown heights (e.g., one was 5 times another). Twelve of the 22 participating students operated with the second multiplicative concept, which meant they viewed known quantities as units of units, or two-levels-of-units structures, but not as three-levels-of-units structures. These students were challenged to represent multiplicative relationships between unknowns: They changed the given relationship, did not think of the relationship as multiplicative until after concerted work, and used numerical values in lieu of unknowns. Our account for these challenges is that students needed to simplify the involved units coordinations. Ultimately students abstracted the relationship as multiplicative, but the exact relationship was not certain or had to be constituted in activity. Implications for teaching are explored.     

Connecting multiplicative thinking and mathematical reasoning in the middle years

Mathematical reasoning and problem solving are recognised as essential 21st century skills. However, international assessments of mathematical literacy suggest these are areas of difficulty for many students. Evidenced-based learning trajectories that identify the key ideas and strategies needed to teach mathematics for understanding and support these important capacities over time are needed to support teachers and curriculum developers so that they do not have to rely solely on mathematics content knowledge. Given this goal and recent evidence to suggest a relationship between the development of multiplicative thinking and mathematical reasoning, this paper explores the processes involved in developing a single, integrated scale for multiplicative thinking and mathematical reasoning using data from a four-year design-based project to establish learning and assessment frameworks for algebraic, geometrical and statistical reasoning in the middle years of schooling.