George Gadanidis

The aesthetic in mathematics as story

Human cognition is story based. We think in terms of stories; we understand the world in terms of stories that we have already understood; we leam by living and accommodating new stories; and we define ourselves through the stories we tell ourselves. In this paper we conceptualize mathematical engagement as aesthetic and place it within the context of mathematics as story. We imagine mathematics teaching and learning experiences as stories acted out on an educational stage. We consider what types of teacher and student roles make good mathematical stories and make a classroom story worth living and discuss the roles the aesthetic may play within such stories. We contend that mathematics is an aesthetic and a storied experience. In this paper we explore the interplay between what is anaesthetic mathematics experience and what is a good mathematics story. We do this in the context of a mathematical applet, namely the Colour Calculator (Sinclair, 2001), and in the coniext of our research into the mathematical and pedagogical thinking of elementary mathematics teachers. In the end, we s.uggest that it is redundant to use both aesthetic and story to describe the mathematics experience, as stories are aesthetic in nature. First, however, we discuss what we mean by aesthetic and by story.RésuméDans le cadre de notre recherche sur les expériences de formation des enseignants de mathématiques — expériences qui aident les enseignants à percevoir les mathématiques et l’enseignement de cette discipline sous un nouveau jour — l’esthétique est présente dans quatre situations: (a) dans la perception qu’ont les enseignants des mathématiques, perception qui ressort lorsqu’ils racontent leurs expériences liées à la discipline; (b) dans les moments les plus marquants de leur formation didactique; (c) dans la façon dont les enseignants de mathématiques réagissent aux expériences les plus riches dans un contexte de formation des enseignants; (d) dans les réactions des enseignants en formation devant l’affection que manifestent les mathématiciens à l’égard de leur discipline (Gadanidis et Hoogland, 2002; Gadanidis, Hoogland et Hill 2002a, 2002b). Nous avons remarqué que les enseignants considèrent l’esthétique comme particulièrement utile lorsqu’il s’agit d’identifier et de décrire les expériences qui ont profondément affecté leur perception des mathématiques et de l’enseignement. D’autre part, nous nous sommes demandé s’il n’existait pas un construit plus vaste capable de lier les aspects esthétiques de l’expérience aux aspects narratifs également évidents dans nos recherches. Par exemple, nous avons remarqué que les enseignants dont la perception des mathématiques est positive racontent souvent qu’ils ont eu des expériences précoces de l’esthétique et des mathématiques dans leur famille ou leur contexte social. Il ressort également que de telles expériences leur servent de modèles lorsqu’il s’agit de structurer leur future conception des mathématiques.La cognition humaine se fonde sur le récit. En effet, l’être humain pense en termes narratifs, comprend le monde en termes des récits qu’il a déjà assimilés, apprend grâce aux récits qu’il vit et qu’il transforme, et se définit grâce aux récits qu’il se raconte. Dans cet article, nous conceptualisons l’engagement en mathématiques comme lui-même esthétique, et nous le situons dans le contexte des mathématiques comme récit. Nous imaginons l’enseignement des mathématiques et les expériences d’apprentissage dans cette discipline comme des récits représentés sur une sorte de scène éducative. Nous analysons quels sont les types de rôles d’enseignants et d’étudiants susceptibles de créer de bons récits mathématiques et d’inventer des récits de classe qui méritent d’être vécus, ainsi que le rôle que peut jouer l’esthétique dans de tels récits. Nous analysons également les conditions dans lesquelles les enseignants peuvent mieux prendre conscience des récits qui caractérisent leurs classes afin d’y réfléchir et de les améliorer. Le fait de concevoir les mathématiques comme récit permet d’ajouter à la discipline un certain nombre de dimensions intéressantes. D’abord, cela permet d’y incorporer naturellement l’esthétique et de lier cet aspect à ce qui constitue un bon récit mathématique. Ensuite, cela ajoute une dimension directionnelle aux mathématiques puisque le récit est vécu dans le temps, et comme direction possible dans la vie. Troisièmement, les mathématiques comme récit sont en mesure d’embrasser la complexité de l’expérience mathématique mieux que ne saurait le faire une liste des caractéristiques ce cette expérience. Quatrièmement, cela contribue à communiquer les directions à prendre en termes de réforme des mathématiques, car on peut ainsi aider les enseignants à mieux prendre conscience des récits qu’ils « créent » pour leurs étudiants. Enfin, l’expérience des enseignants lorsqu’il s’agit d’évaluer la qualité du récit peut servir de guide pour la planification des activités d’apprentissage en mathématiques.

Learning with the Use of the Internet

In this chapter we discuss how the Internet is interacting with mathematics education. After briefly discussing the rise of the Internet and its impact on education, we suggest that it has the potential to disrupt mathematics teaching and learning. Moving far beyond its used as a data resource, we suggest the Internet will provide on-demand access to mathematics knowledge through the collaborative, multimodal and performative affordances of the media that it supports. We note that such affordances will not come to fruition until pedagogical practices have adapted to the rapid pace of this technological change. We conclude by noting that such fundamental change in the teaching of mathematics does have many obstacles, not least that approximately two-thirds of the world’s population does not have sufficient access to the Internet–– and in societies where access is available, access to the Internet often remains limited in classroom settings, particularly for students in low socio-economic areas.

New Media and Online Mathematics Learning for Teachers

In this chapter we offer a case study of an online Mathematics for Teachers course through the lens of four affordances of new media: democratization, multimodality, collaboration and performance. Mathematics, perhaps more so than other school subjects, has traditionally been a subject that people do not talk about outside of classroom settings. However, we demonstrate through the case of the Mathematics for Teachers course that this does not have to be the case. Mathematics, even mathematics that traditionally has been seen as abstract or inaccessible, can be talked about in ways that can engage not only adults but also young children. The affordances of new media can help us rethink and disrupt our existing views of mathematics (for teachers and for students) and of how it might be taught and learned, by (1) blurring teacher/student distinctions and crossing hierarchical curriculum boundaries; (2) communicating mathematics in multimodal ways; (3) seeing mathematics as a collaborative enterprise; and (4) helping us learn how to relate good math stories to classmates and family when asked “What did you do in math today?”

Language Diversity and New Media: Issues of Multimodality and Performance

Language diversity in mathematics education is generally concerned with “multiple language use, learning through second or additional languages, learning through minority or oppressed languages, or through majority or dominant languages” (ICMI Study 21 Discussion document—see Appendix). For example, one issue in language diversity is the hegemonic role played by English in many countries to displace mother tongue education (Kirkpatrick, 2007; Phillipson, 2009). In this chapter, we take up the suggestion made by Marilyn Martin-Jones during the plenary of the ICMI Study 21 conference that future issues on language diversity need to be concerned with the use of digital media, and we explore the relationship between language diversity and new media.

Selection of Apps for Teaching Difficult Mathematics Topics: An Instrument to Evaluate Touch-Screen Tablet and Smartphone Mathematics Apps

Manipulatives—including the more recent touch-screen mobile device apps—belong to a broader network of learning tools . As teachers continue to search for learning materials that aid children to think mathematically, they are faced with a challenge of how to select materials that meet the needs of students. The profusion of virtual learning tools available via the Internet magnifies this challenge. What criteria could teachers use when choosing useful manipulatives? In this chapter, we share an evaluation instrument for teachers to use to evaluate apps . The dimensions of the instrument include: (a) the nature of the curriculum addressed in the app—emergent , adaptable or prescriptive, and relevance to current, high quality curricula —high, medium, low; (b) degree of actions and interactions afforded by the app as a learning tool—constructive, manipulable, or instructive interface; (c) the level of interactivity and range of options offered to the user—multiple or mono, or high, moderate or low; and, (d) the quality of the design features and graphics in the app—rich, high quality or impoverished, poor quality. Using these dimensions, researchers rated the apps on a three-level scale: Levels I, II, and III. Few apps were classified as Level III apps on selected dimensions. This evaluation instrument guides teachers when selecting apps. As well, the evaluation instrument guides developers in going beyond apps that are overly prescriptive, that focus on quizzes, that are text based, and include only surface aspects of using multi-modality in learning, to apps that are more aligned with emergent curricula, that focus also on conceptual understanding, and that utilize multiple, interactive representations of mathematics concepts.

Group Theory, Computational Thinking, and Young Mathematicians

ABSTRACT In this article, we investigate the artistic puzzle of designing mathematics experiences (MEs) to engage young children with ideas of group theory, using a combination of hands-on and computational thinking (CT) tools. We elaborate on: (1) group theory and why we chose it as a context for young mathematicians’ experiences with symmetry and transformations; (2) our ME design principles of agency, access, surprise and audience; (3) the affordances of CT that complement our design principles; and (4) three ME variations we tested in grades 3–6 classrooms. We then reflect on the ME variations based on our design principles and the affordances of CT, and consider how the MEs may be further adapted and improved.

Digital Technology in Mathematics Education: Research over the Last Decade

In this survey paper we focus on identifying recent advances in research on digital technology in the field of mathematics education. We have used Internet search engines with keywords related to mathematics education and digital technology and have reviewed some of the main international journals. We identify five sub-areas of research, important trends of development, and illustrate them using case studies: mobile technologies, massive open online courses (MOOCs), digital libraries and designing learning objects, collaborative learning using digital technology, and teacher training using blended learning. These exemplary case studies may help the reader to understand how recent developments in this area of research have evolved in the last few years. We conclude the report discussing some of the implications that these digital technologies may have for mathematics education research and practice as well as making some recommendations for future research in this area.

The I Teach Mathematics Online Project: Learning and Teaching through Innovative Practices

Providing professional development and support resources that offer additional learning to what teachers might have studied at school, university, and in practice is an increasingly recognized way to support teachers Web-based resources promise to deliver content and pedagogical knowledge in ways enriched by digital technologies. We report on a prototype of a project, I Teach Math project, ITM, developed to deliver pedagogical content knowledge for teaching through problem-solving. ITM was designed from video interviews of selected mathematics teachers on their favorite lessons. On the ITM online database the videos are presented in short clips. Virtual learning objects are used to annotate and illustrate the content. The online environment was harnessed to aid teachers to observe exemplary teaching practices, to build a database of exemplary teaching, and to share ideas on teaching practices. In the process of designing ITM we surveyed existing online projects to select 10 major players for a comparative analysis. This paper reports on the development of ITM. It explores the digital-technologies utilized, the pedagogical content knowledge and pedagogical thinking shared by the teachers.

Teacher Tasks for Mathematical Insight and Reorganization of What it Means to Learn Mathematics

The mathematics-for-teachers tasks we discuss in this chapter have two qualities: (1) they offer teachers opportunities to experience the pleasure of mathematical insight; and (2) they aim to disrupt and reorganize teachers’ views of what it means to do and learn mathematics. Given that many future and inservice elementary teachers fear and dislike mathematics, it is perhaps not too far-fetched to suggest that there is a need for “math therapy.” We believe that a form of mathematics therapy may involve new and different experiences with mathematics. Such experiences, considered broadly to include questions or prompts for mathematical exploration, draw attention to deep mathematical ideas and offer the potential of experiencing the pleasure of significant mathematical insight. In our work with teachers we have developed and used a variety of mathematics tasks as opportunities for experiential therapy. The tasks aim to challenge some of the mathematical myths that future teachers believe to be true and are typically assumed by them in mathematics classrooms. The tasks have potential to disrupt teachers’ view of mathematics, and to start the process for reorganizing their thinking about what mathematics is and what it means to do and learn mathematics. In this chapter we describe and discuss four of the mathematics tasks which involve non-routine mathematics problems that we use in our mathematics-forteachers program. This program is offered annually to our 440 future elementary school (K-8) teachers, who generally lack confidence in mathematics and often fear and/or dislike the subject. It is also offered to inservice teachers through a series of mathematics-for-teachers courses. A student response summarizes the effects of our approach.