Leading students towards the formal world of mathematical thinking: a mathematician’s reflections on teaching eigentheory

ABSTRACT

In this study, we analysed a mathematician’s teaching journals on eigenvalues and eigenvectors in a first-year linear algebra course. The research team employed Tall’s [How humans learn to think mathematically: Exploring the three worlds of mathematics. Cambridge University Press] three-world model of embodied, symbolic and formal as a framework for understanding the mathematician and teacher’s pedagogical reflections as he led the class to the formal world. In order to reach the formal world, he used a sequence of tasks that emphasized embodied and symbolic, as well as formal thinking. The analysis of the journals showed that the mathematician faced challenges in leading the class towards the formal world. The study also revealed that the mathematician strived to build a concept image, that, while perhaps mirroring his own, did not resonate with the students.     

Shifts of mathematical thinking in adolescence

This theoretical paper relates key features of the mathematics adolescents are expected to learn in school to other aspects of adolescent development. Difficulties in mathematical learning at that age include changes in perspective and in the actions that are mathematically productive. Commonly-recommended methods of trying to engage adolescents in mathematics do not necessarily enable students to shift to new perceptions and new ways of constructing mathematical understandings, yet the shifts students need to make are in accord with other aspects of adolescent development.     

Teachers’ mathematical activity in inquiry-oriented instruction

This work investigates the relationship between teachers’ mathematical activity and the mathematical activity of their students. By analyzing the classroom video data of mathematicians implementing an inquiry-oriented abstract algebra curriculum I was able to identify a variety of ways in which teachers engaged in mathematical activity in response to the mathematical activity of their students. Further, my analysis considered the interactions between teachers’ mathematical activity and the mathematical activity of their students. This analysis suggests that teachers’ mathematical activity can play a significant role in supporting students’ mathematical development, in that it has the potential to both support students’ mathematical activity and influence the mathematical discourse of the classroom community.     

工科高校《应用近世代数》课程体系改革探讨与思考

文章从阐述近世代数在工科高校教学中的重要意义及对现有教材存在的一些问题的探讨出发,从几个方面考察了目前工科高校《应用近世代数》课程体系教学改革的必要性,并提出了自己的一些设想,期望能为工科高校的《应用近世代数》教学改革及高校教育改革抛砖引玉.     

Instances of mathematical thinking among low attaining students in an ordinary secondary classroom

This paper is a report of a classroom research project whose aim was to find out whether low attaining 14-year-old students of mathematics would be able to think mathematically at a level higher than recall and reproduction during their ordinary classroom mathematics activities. Analysis of classroom interactive episodes revealed many instances of mathematical thinking of a kind which was not normally exploited, required or expected in their classes. Five episodes are described, comparing the students’ thinking to that usually described as “advanced.” In particular, some episodes suggest the power of a type of prompt which can be generalized as “going across the grain.”     

Remarks on the rank properties of formal CR maps

We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

The relationship between students' mathematical thinking types and representation preferences in definite integral problems

ABSTRACT

Given that curriculum materials serve as cultural artefacts, this study addresses the need for more research on curriculum materials in different contexts. Most studies concerning curriculum materials have been conducted in US and, therefore little is known about the nature of materials in other cultural-educational contexts. The aim of this paper is to identify the underlying cultural norms of potentially constructed classrooms, by analysing recurrent activities in the most commonly used Finnish teacher guides at primary-school level. We identified three norms embedded in them: (1) creating opportunities for learning through a variety of activities and communication; (2) keeping the class gathered around a specific mathematical topic; and (3) concurrent active involvement of teachers and students. The results add to knowledge about both teacher guides and the Finnish educational context. Moreover, it adds to the growing body of methodologies, as our analytical approach is novel in the context of textual analysis.     

On the Gevrey order of formal solutions of nonlinear difference equations

We prove a Maillet type theorem for formal solutions of nonlinear difference systems, relating the Gevrey order of the formal solutions to the lowest level of an associated, linear difference operator.     

A constructive proof of the Peter‐Weyl theorem

We present a new and constructive proof of the Peter‐Weyl theorem on the representations of compact groups. We use the Gelfand representation theorem for commutative C*‐algebras to give a proof which may be seen as a direct generalization of Burnside's algorithm [3]. This algorithm computes the characters of a finite group. We use this proof as a basis for a constructive proof in the style of Bishop. In fact, the present theory of compact groups may be seen as a natural continuation in the line of Bishop's work on locally compact, but Abelian, groups [2]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Refinement of Ado's Theorem in Low Dimensions and Application in Affine Geometry

In this article, we construct a faithful representation with the lowest dimension for every complex Lie algebra in dimension ≤ 4. In particular, in our construction, in the case that the faithful representation has the same dimension of the Lie algebra, it can induce an étale affine representation with base zero which has a natural and simple form and gives a compatible left-symmetric algebra on the Lie algebra. Such affine representations do not contain any nontrivial one-parameter subgroups of translation.     

Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials

In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp (g) for some exponential polynomial g, then p and q have only finitely many common zeros.     

Invariant hermitian lattices in the steinberg module and their isometry groups

The invariant Hermitian lattices in the Steinberg module of SL₂(q) are described. These lattices are connected with generalized quadratic residue codes over a field of four elements. The isometry groups of in-variant lattices are calculated. In particular, lbdimensional unimodular lattices over Eisenstein numbers with minimum norm 3 and automor- phism group Z₆x PSp₆(3) are obtained.     

Symmetry of cuts in fields of formal power series

The theory of cuts is an effective tool for studying ordered fields. We continue research into the relationship between the structure of cuts in a field of formal power series and algebraic properties of the field. __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 174–185, March–April, 2008.     

Applications of Category Theory to the Area of Algebraic Specification in Computer Science

The theory of algebraic specifications – one of the most important mathematical approaches to the specification of abstract data types and software systems – is reviewed from a mathematical and a computer science point of view. The important role of category theory in this area is discussed and it is shown how the following selected problems are treated using category theory: First, a unified framework for specification logics, second compositional semantics, third partial algebras and their specification, and fourth specifications and models for concurrent systems. For the solution of two of the problems classifying categories are used. They allow to present categories of algebras as functor categories and to derive a number of important properties from well known results for functor categories.     

The potential of recreational mathematics to support the development of mathematical learning

ABSTRACT

A literature review establishes a working definition of recreational mathematics: a type of play which is enjoyable and requires mathematical thinking or skills to engage with. Typically, it is accessible to a wide range of people and can be effectively used to motivate engagement with and develop understanding of mathematical ideas or concepts. Recreational mathematics can be used in education for engagement and to develop mathematical skills, to maintain interest during procedural practice and to challenge and stretch students. It can also make cross-curricular links, including to history of mathematics. In undergraduate study, it can be used for engagement within standard curricula and for extra-curricular interest. Beyond this, there are opportunities to develop important graduate-level skills in problem-solving and communication. The development of a module ‘Game Theory and Recreational Mathematics’ is discussed. This provides an opportunity for fun and play, while developing graduate skills. It teaches some combinatorics, graph theory, game theory and algorithms/complexity, as well as scaffolding a Pólya-style problem-solving process. Assessment of problem-solving as a process via examination is outlined. Student feedback gives some indication that students appreciate the aims of the module, benefit from the explicit focus on problem-solving and understand the active nature of the learning.     

Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning

The purpose of this paper is to further the notion of defining as a mathematical activity by elaborating a framework that structures the role of defining in student progress from informal to more formal ways of reasoning. The framework is the result of a retrospective account of a significant learning experience that occurred in an undergraduate geometry course. The framework integrates the instructional design theory of Realistic Mathematics Education (RME) and distinctions between concept image and concept definition and offers other researchers and instructional designers a structured way to analyze or plan for the role of defining in students’ mathematical progress.