Reductive-holistic cycle: A model for the study of the didactic procedure

The concept ofreality level may be useful as a catalyst among several systems in the area of knowledge. This concept is leading us to ask, if we can make a reduction from a reality level to another, that is to the problem ofreductionism. Relative to it is the problem ofholism. At the end these concepts are connected to the category theory and adjoint functors. Within the framework of this aspect we set up a model for the study of the didactic procedure. This model is a feedback system between two reality levels or categories, these of the teacher and of the student. So, the article seeks to enhance and improve the teaching of mathematics by its attempt to understand both student's and teacher's knowledge in the same terms.     

How Preservice Teachers Interpret and Respond to Student Geometric Errors

Recognizing and responding to students' thinking is essential in teaching mathematics, especially when students provide incorrect solutions. This study examined, through a teaching scenario task, elementary preservice teachers' interpretations of and responses to a student's work on a task involving reflective symmetry. Findings revealed that a majority of preservice teachers identified the student's errors from conceptual aspects of reflection rather than from procedural aspects. However, when they responded to the student's errors, preservice teachers tried to cope with them by invoking procedural knowledge. This study also revealed the three types of responses and two different forms of address by preservice teachers to student errors; these categories might provide insight into the difficulties arising in communication between students and teachers.     

Extension of Henrici's method to matrix sequences

In this paper we generalize the definition of linear convergence to matrix sequences. This new definition is used to establish some new results useful to study the new extension of Henrici's method. A convergence theorem, an algorithm for implementation of this method and some numerical examples are given.     

Defining a mathematical research school: the case of algebra at the University of Chicago, 1892–1945

Historians of science have long considered the concept of the “research school” as a potent analytical construct for understanding the development of the laboratory sciences. Unfortunately, their definitions fall short in the case of mathematics. Here, a definition of “mathematical research school” is proposed in the context of a case study of algebraic work associated with the University of Chicago's Department of Mathematics from the University's founding in 1892 through 1945.     

Different aspects of the monotonicity of a function

In this paper, we investigate which aspects are overriding in the concept images of monotonicity of Finnish tertiary mathematics students, i.e., on which aspects of monotonicity they base their argument in different types of exercises related to that concept. Further, we examine the relationship between the quality of principal aspects and the success in solving monotonicity exercises and a few other standard problems in calculus. Our findings indicate that a mathematics student's conception about monotone functions is often restricted to continuous or differentiable functions and the algebraic aspect – the nearest one to the formal definition of monotonicity – is rare.     

The Impact on Incorporating Collaborative Concept Mapping with Coteaching Techniques in Elementary Science Classes

The purpose of this research was to evaluate a collaborative concept‐mapping technique that was integrated into coteaching in fourth‐grade science classes in order to examine students' performance and attitudes toward the experimental teaching method. There are two fourth‐grade science teachers and four classes with a total of 114 students involved in the study. This study used a mixed method design, incorporating both quantitative and qualitative techniques. The findings showed that the two teaching methods obtained significant difference with respect to students' test scores. Using collaborative concept mapping to learn science could increase the opportunity of discussion between peers, thus fostering better organization and understanding the content. In addition, coteaching could enable teachers to share their expertise with one another. It could facilitate the implementation of collaborative concept mapping and the construction of student's concept mapping. Team teachers' attitude could affect the students' learning performance. However, some of the students had negative views on drawing concept maps because they found it was troublesome to write down many words, difficult to draw and arrange proposition, and time‐consuming. Coteachers' instant feedback and students' journal writing could guide and examine the students' concept maps to facilitate their cognitive learning.     

On the Kolmogorov complexity of continuous real functions

Kolmogorov complexity was originally defined for finitely-representable objects. Later, the definition was extended to real numbers based on the asymptotic behaviour of the sequence of the Kolmogorov complexities of the finitely-representable objects—such as rational numbers—used to approximate them.This idea will be taken further here by extending the definition to continuous functions over real numbers, based on the fact that every continuous real function can be represented as the limit of a sequence of finitely-representable enclosures, such as polynomials with rational coefficients.Based on this definition, we will prove that for any growth rate imaginable, there are real functions whose Kolmogorov complexities have higher growth rates. In fact, using the concept of prevalence, we will prove that ‘almost every’ continuous real function has such a high-growth Kolmogorov complexity. An asymptotic bound on the Kolmogorov complexities of total single-valued computable real functions will be presented as well.     

NEEDING TO USE ALGEBRA

We have completed a Teacher Training Agency (TTA) funded project looking at the teaching and learning of algebra with one mixed-ability year 7 class (11–12 year-olds in the UK). We take Kieran's (quoted in Sutherland, 1997) definition of algebra and see ourselves as developing a'community of practice’ (Lave and Wenger, 1991) in the classroom where the practice is not that of mathematician but of inquirer (Schoenfeld, 1996) into mathematics. The teacher in this community acts as role model for inquirer and metacomments (Bateson, 1972) on the practice of inquiry. In this paper we present evidence for how such metacommenting supports the development of a'community of inquiry’ which leads to the'need’ for algebraic activity. We conclude with a case study, illustrating one student's need for algebra.     

On computing the entropy of braids

We consider the problem of computing the entropy of a braid. We recall its definition and for each braid construct a sequence of real numbers whose limit is the braid’s entropy. We state one conjecture on the convergence speed and two conjectures on braids that have high entropy but are written with few letters.      

Normalized convergence in stochastic optimization

A new concept of (normalized) convergence of random variables is introduced. This convergence is preserved under Lipschitz transformations, follows from convergence in mean and itself implies convergence in probability. If a sequence of random variables satisfies a limit theorem then it is a normalized convergent sequence. The introduced concept is applied to the convergence rate study of a statistical approach in stochastic optimization.     

Mathematics education graduate students’ understanding of trigonometric ratios

This study describes mathematics education graduate students’ understanding of relationships between sine and cosine of two base angles in a right triangle. To explore students’ understanding of these relationships, an elaboration of Skemp's views of instrumental and relational understanding using Tall and Vinner's concept image and concept definition was developed. Nine students volunteered to complete three paper and pencil tasks designed to elicit evidence of understanding and three students among these nine students volunteered for semi-structured interviews. As a result of fine-grained analysis of the students’ responses to the tasks, the evidence of concept image and concept definition as well as instrumental and relational understanding of trigonometric ratios was found. The unit circle and a right triangle were identified as students’ concept images, and the mnemonic was determined as their concept definition for trigonometry, specifically for trigonometric ratios. It is also suggested that students had instrumental understanding of trigonometric ratios while they were less flexible to act on trigonometric ratio tasks and had limited relational understanding. Additionally, the results indicate that graduate students’ understanding of the concept of angle mediated their understanding of trigonometry, specifically trigonometric ratios.     

Increasing the applicability of Steffensen's method

We present an original alternative to the majorant principle of Kantorovich to study the semilocal convergence of Steffensen's method when it is applied to solve nonlinear systems which are differentiable. This alternative allows choosing starting points from which the convergence of Steffensen's method is guaranteed, but it is not from the majorant principle. Moreover, this study extends the applicability of Steffensen's method to the solution of nonlinear systems which are nondifferentiable and improves a previous result given by the authors.     

关于叶果洛夫定理证明的一个注记

给出了一种函数列收敛的等价刻画,并利用此等价刻画来给出实变函数中叶果洛夫定理的另一种证明方法.     

Mastering by the teacher of the instrumental genesis in CAS environments: necessity of intrumental orchestrations

In this article, we study didactic phenomena identified in integration experiments within our classes, CAS (implemented in calculators). From this study, we show the interest of an instrumental approach to understand the influence of tools on the mathematical approach and on the building of student's knowledge: through a process—instrumental genesis—a calculator becomes a mathematical work tool; this process depends on the tool's constraints and potentialities, on students' knowledge, and on the class' work situations. To analyze the differentiation of instrumental genesis, we then have taken interest in students' behaviour and we propose a method enabling us to constitute a typology of extreme behaviour in environments of symbolic calculators. To take the variety of these genesis into account, the professor needs a particular organization of space and time of the study in the class. We suggest the notion of instrumental orchestration to name this organization. We demonstrate how this notion gives a better definition of the objectives, the configurations and the exploitation modes of different arrangements which aim at constituting coherent instrument systems for each student and for the class.     

Independence and convergence in non-additive settings

Some properties of convergence for archimedean t-conorms and t-norms are investigated and a definition of independence for events, evaluated by a decomposable measure, is introduced. This definition generalizes the concept of independence provided by Kruse and Qiang for λ-additive fuzzy measures. Finally, we derive the two Borel–Cantelli lemmas in the context of the general framework considered.     

Immigrant parents’ perspectives on their children’s mathematics education

This paper draws on two research studies with similar theoretical backgrounds, in two different settings, Barcelona (Spain) and Tucson (USA). From a sociocultural perspective, the analysis of mathematics education in multilingual and multiethnic classrooms requires us to consider contexts, such as the family context, that have an influence on these classrooms and its participants. We focus on immigrant parents' perspectives on their children's mathematics education and we primarily discuss two topics (1) their experiences with the teaching of mathematics, and (2) the role of language (native language and second language). The two topics are explored with reference to the immigrant student's or their parents' former educational systems (the “before”) and their current educational systems (the “now”). Parents and schools understand educational systems, classroom cultures and students' attainment differently, as influenced by their sociocultural histories and contexts.     

Factors Influencing Prospective Teachers’ Recommendations to Students: Horizons,Hexagons, and Heed

This article examines pre-service secondary school teachers’ responses to a learning situation that presented a student's struggle with determining the area of an irregular hexagon. Responses were analyzed in terms of participants’ evoked concept images as related to their knowledge at the mathematical horizon, with attention paid toward the influence of one on the other. Specifically, our analysis attends to common features in participants’ understanding of the mathematical task, and explores the interplay between participants’ personal solving strategies and approaches and their identified preferences when advising a student. We conclude with implications for mathematics teacher education research and pedagogy.     

A generalization of almost convergence

In this paper, we discuss almost convergent sequences. The concept of almost convergence was first introduced by G. G. Lorentz in 1948. Since then, various types of generalizations for almost convergence have been constructed and studied. Our concern is also to generalize the almost convergence. We construct a generalized almost convergence (GAC) in a natural way through the use of Lorentz-type definition, and give an analogue of Lorentz’s result. It should be noted that the definition of our GAC does not require normed space nor boundedness of sequences, although Lorentz-type definitions usually require it. On the other hand, we also show that our GAC eventually has a certain type of boundedness in spite of its definition.