On affine images of regular pentagons and pentagrams

In this paper we explore pentagons that are affine images of the regular pentagon and the regular pentagram. We obtain their characterizations in terms of two mild forms of regularity that deal with the notions of medians for a pentagon and the natural requirement that they are concurrent. Using these characterizations we show that there are various values involving the number 5 (thus related to the golden section) for which a careful selection of division points on appropriate segments determined by any pentagon will result in a pentagon that is the affine image of either a regular pentagon or a regular pentagram.     

Sub(L)中的五边形格特征及其数量不等式

五边形结构在刻画格的特征方面具有十分重要作用，应用格论及组合数学的方法讨论了格L与其子格格Sub(L)中所含五边形格之间的数量关系，给出了有限格L的子格格中三个元生成五边形格的充要条件，同时给出了Sub(L)所含不同五边形格数量的一个下界。     

Research, practice, uncertainty and responsibility

Three issues concerning the relationship between research and practice are addressed. (1) A certain ‘prototype mathematics classroom’ seems to dominate the research field, which in many cases seems selective with respect to what practices to address. I suggest challenging the dominance of the discourse created around the prototype mathematics classroom. (2) I find it important to broaden the school-centred discourse on mathematics education and to address the very different out-of-school practices that include mathematics. Many of these practices are relevant for interpreting what is taking place in a school context. That brings us to (3) socio-political issues of mathematics education. When the different school-sites for learning mathematics as well as the many different practices that include mathematics are related, we enter the socio-political dimension of mathematics education.On the one hand we must consider questions like: Could socio-political discrimination be acted out through mathematics education? Could mathematics education exercise a regimentation and disciplining of students? Could it include discrimination in terms of language? Could it include sexism and racism? On the other hand: Could mathematics education bring about competencies which can be described as empowering, and as supporting the development of mathematical literary or a ‘mathemacy’, important for the development of critical citizenship?However, there is no hope for identifying a one-way route to mathemacy. More generally: There is no simple way of identifying the socio-political functions of mathematics education. Mathematics education has to face uncertainty, and this challenge brings us to the notion of responsibility.     

Extra two-fold Steiner pentagon systems

A two-fold pentagon system is a decomposition of the complete 2-multigraph (every two distinct vertices joined by two edges) into pentagons. A two-fold Steiner pentagon system is a two-fold pentagon system such that every pair of distinct vertices is joined by a path of length two in exactly two pentagons of the system. We consider two-fold Steiner pentagon systems with an additional property : for any two vertices, the two paths of length two joining them are distinct. We determine completely the spectrum for such systems, and point out an application of such systems to certain 4-cycle systems.     

Soap films and GeoGebra in the study of Fermat and Steiner points

We discuss how mathematics and secondary mathematics education majors developed an understanding of Fermat points for the triangle as well as Steiner points for the square and regular pentagon, and also of soap film configurations between parallel plates where forces are in equilibrium. The activities included the use of soap films and the interactive geometry program GeoGebra. Students worked in small groups using these tools to investigate the properties of Fermat and Steiner points and then justified the results of their investigations using geometrical arguments. These activities are specific approaches of how to encourage prospective teachers to use physical experiments to support students’ development of mathematical curiosity and mathematical justifications.     

Unavoidable Crossings in a Thinnest Plane Covering with Congruent Convex Disks

Two convex disks K and L in the plane are said to cross each other if the removal of their intersection causes each disk to fall into disjoint components. Almost all major theorems concerning the covering density of a convex disk were proved only for crossing-free coverings. This includes the classical theorem of L. Fejes Tóth (Acta Sci. Math. Szeged 12/A:62–67, 1950) that uses the maximum area hexagon inscribed in the disk to give a significant lower bound for the covering density of the disk. From the early seventies, all attempts of generalizing this theorem were based on the common belief that crossings in a plane covering by congruent convex disks, being counterproductive for producing low density, are always avoidable. Partial success was achieved not long ago, first for “fat” ellipses (A. Heppes in Discrete Comput. Geom. 29:477–481, 2003) and then for “fat” convex disks (G. Fejes Tóth in Discrete Comput. Geom. 34(1):129–141, 2005), where “fat” means of shape sufficiently close to a circle. A recently constructed example will be presented here, showing that, in general, all such attempts must fail. Three perpendiculars drawn from the center of a regular hexagon to its three nonadjacent sides partition the hexagon into three congruent pentagons. Obviously, the plane can be tiled by such pentagons. But a slight modification produces a (non-tiling) pentagon with an unexpected covering property: every thinnest covering of the plane by congruent copies of the modified pentagon must contain crossing pairs. The example has no bearing on the validity of Fejes Tóth’s bound in general, but it shows that any prospective proof must take into consideration the existence of unavoidable crossings.     

Policy and the standards debate: mapping changes in assessment in mathematics

ABSTRACT

The influences on governments for policy changes in schools range across many agencies, including the political party in power. When policies change, the sources of these influences are not always clear. The project whose work is presented in this special issue examines what these changes look like in terms of the differences in assessment tasks of school pupils’ mathematics, over time. In this article we attempt to develop a graph, which we argue will have general applicability internationally, that can help to reveal the sources and nature of those influences. We construct the graph in interaction with an examination of the most recent changes in two countries. We argue that our analysis is a necessary complement to the project’s findings in that it enables us to identify the fields of recontextualisation, their relative strengths in terms of influence and hence conjecture their impact on the mathematics curriculum.     

A framework and rubric for guiding the training of mathematics tutors in third-level education

This paper uses a framework established by one of the author's in a previous study and refines it to develop a rubric for mathematics tutor training in third-level education. The paper further opens the discussion on the necessity of, and practical methods for, providing high quality, efficient tutor training at third-level. Models of mathematics teacher knowledge are unanimous in stating that solely possessing strong subject matter knowledge (SMK) is not sufficient for one to be deemed to be a good teacher. Despite the necessity of high quality tutoring at third-level, past research indicates that though tutors’ SMK is usually of high quality, their ability to transform their knowledge so that students will understand the content, as well as their general teaching skills (e.g. board work, speech), often has room to improve. This paper discusses the development of the rubric and its use in a mathematics tutor training programme at third-level. The tutor training programme was found to lead to improvements in multiple elements of the tutors’ teaching.     

高等数学教学现状调查分析

高等学校扩大招生后,非数学专业的数学教育面临很多问题,大部分人把这些问题归结为学生质量的下降造成的,这是不公正的.教学过程是一个系统的过程,从对1995级和2005级同一份高等数学(下册)试卷的考试情况进行分析,虽然可以显现出学生的差异,但这些差异不全是学生的质量的问题,应该与教学环境、教学要求、教学方法、师资条件以及社会发展环境等相关联,同时与学生的学习兴趣、学习能力是密不可分的.调查分析这些因素对当代学生的学习和数学教学的影响程度到底有多大,是有效提高大学数学教学质量的前提.     

Difference and differential equations in the mathematics of finance

Many problems arising in the mathematics of finance involve identical money flows at regular time intervals and are typically solved by appropriate valuation at a focal date or by setting up an equation of value. It is shown here that such problems can be viewed as special cases of a certain class of first‐order difference equations. Problems relating to continuous money flows can be viewed analogously as special cases of a certain class of ordinary first‐order differential equations. Students should thus be encouraged to view the concepts and techniques of the mathematics of finance as not being inherently different from those prevalent in more traditional applied mathematics.     

Orthogonal triangulation of polygons

In this study, the question whether any convex polygon can be divided using an orthogonal grid into right-angled triangles is answered in the negative. Moreover, it is demonstrated that there exists a convex pentagon which cannot be even approximated by divisible pentagons.     

数学思想方法在高等数学教育中的作用

从高等数学教育改革的角度,阐述了数学思想方法的含义和高等数学中的基本数学思想方法,论述了在高等数学教育中加强数学思想方法教学的重要性.     

Generalising compactness

Working within the framework of Bishop's constructive mathematics, we will show that it is possible to define compactness in a more general setting than that of uniform spaces. It is also shown that it is not possible to do this in a topological space. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

大学数学与中学数学教学内容衔接研究

数学教育是一个完整的科学体系,对大学数学教育与中学数学教育进行衔接研究具有重要意义,而教学内容的衔接是二者衔接的一个重要方面.本文根据大学数学和中学数学教学中较普遍使用的经典教材,对大学数学和中学数学教学内容从数学符号使用、教学内容重叠、教学内容遗漏三个方面进行详细归纳总结,对于大学数学教学和大学数学教材的编写具有一定借鉴作用.     

Reinventing solutions to systems of linear differential equations: A case of emergent models involving analytic expressions

An enduring challenge in mathematics education is to create learning environments in which students generate, refine, and extend their intuitive and informal ways of reasoning to more sophisticated and formal ways of reasoning. Pressing concerns for research, therefore, are to detail students’ progressively sophisticated ways of reasoning and instructional design heuristics that can facilitate this process. In this article we analyze the case of student reasoning with analytic expressions as they reinvent solutions to systems of two differential equations. The significance of this work is twofold: it includes an elaboration of the Realistic Mathematics Education instructional design heuristic of emergent models to the undergraduate setting in which symbolic expressions play a prominent role, and it offers teachers insight into student thinking by highlighting qualitatively different ways that students reason proportionally in relation to this instructional design heuristic.     

新课标体系下高中数学对大学工科数学教学产生的问题分析及对策探索

大学工科数学的部分基础内容下移于高中数学,高中数学中原有的基础内容又有所删减,大学工科数学的教材内容并没有因此而调整.所以,新课标体系下的高中数学教学对大学工科数学教学产生了深刻的影响.本文分析了大学工科数学教学面临的问题,在此基础上,结合作者多年的教学实践,探索性的提出了解决问题的对策.     

On a Characterization Theorem of Semimodular Lattices

In general lattice theory,there are two basic criterion theorems:　 (1 ) A lattice L is modular if and only if L does not contain a pentagon N5(seeFig. 1 a) .　 (2 ) A lattice L is distributive ifand only if L contains neither a pentagon N5nor a dia-mond M3(see Fig. 1 b) .　 Now,the problem is that if L is semimodular,whether L possesses the similar char-acteristic?In this regard,the following conclusion is obtained in this paper:　 A lattice L is semimodular if and only if L does not co…     

A case for combinatorics: A research commentary

In this commentary, we make a case for the explicit inclusion of combinatorial topics in mathematics curricula, where it is currently essentially absent. We suggest ways in which researchers might inform the field’s understanding of combinatorics and its potential role in curricula. We reflect on five decades of research that has been conducted since a call by Kapur (1970) for a greater focus on combinatorics in mathematics education. Specifically, we discuss the following five assertions: 1) Combinatorics is accessible, 2) Combinatorics problems provide opportunities for rich mathematical thinking, 3) Combinatorics fosters desirable mathematical practices, 4) Combinatorics can contribute positively to issues of equity in mathematics education, and 5) Combinatorics is a natural domain in which to examine and develop computational thinking and activity. Ultimately, we make a case for the valuable and unique ways in which combinatorics might effectively be leveraged within K-16 curricula.     

Convex pentagons that admit <Emphasis Type="Italic">i</Emphasis>-block transitive tilings

The problem of classifying the convex pentagons that admit tilings of the plane is a long-standing unsolved problem. Previous to this article, there were 14 known distinct kinds of convex pentagons that admit tilings of the plane. Five of these types admit tile-transitive tilings (i.e. there is a single transitivity class with respect to the symmetry group of the tiling). The remaining 9 types do not admit tile-transitive tilings, but do admit either 2-block transitive tilings or 3-block transitive tilings; these are tilings comprised of clusters of 2 or 3 pentagons such that these clusters form tile-2-transitive or tile-3-transitive tilings. In this article, we present some combinatorial results concerning pentagons that admit i-block transitive tilings for \(i \in \mathbb {N}\). These results form the basis for an automated approach to finding all pentagons that admit i-block transitive tilings for each \(i \in \mathbb {N}\). We will present the methods of this algorithm and the results of the computer searches so far, which includes a complete classification of all pentagons admitting i-block transitive tilings for \(i \le 4\), among which is a new 15th type of convex pentagon that admits a tile-3-transitive tiling.     

Extracting factors for students’ motivation in studying mathematics

The purpose of this article is to identify factors that statistically explain the variation and the measures on the level of motivation of a sample of mathematics students in a university. Specifically, this analysis will identify groups of similar items and reduce the number of variables used in a study. This article explains the use of exploratory factor analysis in extracting factors of personal belief and motivational factors among students in learning mathematics. The adaptation of these factors can be used for assessing academic performance in relation to motivation level. By identifying these factors, the mathematics educators or researchers will be able to find ways to improve the condition of the factors and also to further investigate the factors based on confirmatory approaches.