Bharath Sriraman

A global survey of international perspectives on modelling in mathematics education

In this article we survey the current debate on modelling and, describe different perspectives on this debate. We relate these perspectives with earlier perspectives and show similarities and differences between these different approaches.

Are Giftedness and Creativity Synonyms in Mathematics

At the K-12 level one assumes that mathematically gifted students identified by out-of-level testing are also creative in their work. In professional mathematics, “creative” mathematicians constitute a very small subset within the field. At this level, mathematical giftedness does not necessarily imply mathematical creativity but the converse is certainly true. In the domain of mathematics, are the words creativity and giftedness synonyms? In this article, the constructs of mathematical creativity and mathematical giftedness are developed via a synthesis and analysis of the general literature on creativity and giftedness. The notions of creativity and giftedness at the K-12 and professional levels are compared and contrasted to develop principles and models that theoretically “maximize” the compatibility of these constructs. The relevance of these models is discussed with practical considerations for the classroom. The paper also significantly extends ideas presented by Usiskin (2000).

Theories of Mathematics Education

The purpose of this Forum is to stimulate critical debate in the area of theory use and theory development, and to consider future directions for the advancement of our discipline. The Forum opens with a discussion of why theories are essential to the work of mathematics educators and addresses the possible reasons for why some researchers either ignore or misunderstand/misuse theory in their work. Other issues to be addressed include the social turn in mathematics education, an evolutionary perspective on the nature of human cognition, the use of theory to advance our understanding of student cognitive development, and models and modelling perspectives. The final paper takes a critical survey of European mathematics didactics traditions, particularly those in Germany and compares these to historical trends in other parts of the world.

Problem Solving for the 21st Century

Mathematical problem solving has been the subject of substantial and often controversial research for several decades. We use the term, problem solving, here in a broad sense to cover a range of activities that challenge and extend one’s thinking. In this chapter, we initially present a sketch of past decades of research on mathematical problem solving and its impact on the mathematics curriculum. We then consider some of the factors that have limited previous research on problem solving. In the remainder of the chapter we address some ways in which we might advance the fields of problem-solving research and curriculum development.

Mathematical Giftedness, Problem Solving, and the Ability To Formulate Generalizations: The Problem-Solving Experiences of Four Gifted Students.

Complex mathematical tasks such as problem solving are an ideal way to provide students opportunities to develop higher order mathematical processes such as representation, abstraction, and generalization. In this study, 9 freshmen in a ninth-grade accelerated algebra class were asked to solve five nonroutine combinatorial problems in their journals. The problems were assigned over the course of 3 months at increasing levels of complexity. The generality that characterized the solutions of the 5 problems was the pigeonhole (Dirichlet) principle. The 4 mathematically gifted students were successful in discovering and verbalizing the generality that characterized the solutions of the 5 problems, whereas the 5 nongifted students were unable to discover the hidden generality. This validates the hypothesis that there exists a relationship between mathematical giftedness, problem-solving ability, and the ability to generalize. This paper describes the problem-solving experiences of the mathematically gifted students and how they formulated abstractions and generalizations, with implications for acceleration and the need for differentiation in the secondary mathematics classroom.

Towards a didactical theory for mathematical modelling

Mathematics education research has been somewhat short of its own paradigmatic theories (see ZDM, 2005, issue 6 and ZDM 2006, issue 1 for an ongoing discussion). Theories are often borrowed from the background sciences and applied to the field of mathematics education e.g. general learning theories from pedagogy, sociology, psychology etc. This has led to our field being a subject to a host of criticisms (e,g., Lerman, 2006; Lester, 2005; Sriraman & English, 2005, 2006; Steen, 1999). Therefore it is relevant to look for areas in mathematics education where theories may emerge from studying the processes of teaching and learning mathematics. In this issue of ZDM we present and discuss what we consider to be an example of a developing theory within the field of mathematics education research, namely a theory for the teaching and learning of mathematical modelling. During the last decades a coherent theoretical understanding of mathematical modelling processes and connected teaching and learning processes has been emerging. This has happened through a close interplay between the development of curricula, teaching practices, experimental teaching, theoretical reflections and research. Together with the use of information technology, the introduction of mathematical modelling and applications is a prominent general feature of the recent developments in the practice of mathematics teaching, especially with regard to secondary level teaching. In this development, didactical research has already played an important role and we find it of general interest for mathematics education research to examine the development of a theory for mathematical modelling as a paradigmatic case for developing theory in close interplay with teaching practice.

An Exploratory Study of Relationships between Students’ Creativity and Mathematical Problem-Posing Abilities

The literature is replete with statements alleging that people in the Western countries are more creative than people in the East Asian countries (Zhao, 2008; Rudowicz & Hui, 1998; Yue & Rudowicz, 2002; Lubart, 1990; Dunn, Zhang, & Ripple, 1988). A typical explanation for those statements is that, citizens of the Western cultures tend to be independent and to find meaning largely by reference to their own internal thoughts, feelings, and actions rather than by those of others; while citizens of the East tend to hold an interdependent perspective of the self in which meaning depends more on interpersonal relationships (Markus & Kitayama, 1991). In other words, in the Western countries, individuals often focus on discovering and expressing themselves and on accentuating differences from others, whereas East Asians tend to organize more into hierarchies in which individuals seek membership in larger communities (Zha, Walczyk, Grifffith-Ross, & Tobacyk, 2006).

Confronting Dogmatism in Gifted Education

Will reading habit influence your life? Many say yes. Reading confronting dogmatism in gifted education is a good habit; you can develop this habit to be such interesting way. Yeah, reading habit will not only make you have any favourite activity. It will be one of guidance of your life. When reading has become a habit, you will not make it as disturbing activities or as boring activity. You can gain many benefits and importances of reading.

Modeling conceptions revisited

The previous issue of ZDM raised several fundamental issues on the role of modeling in the school curricula at micro and macro levels. In this paper we complement the approaches described there by discussing some of the issues and the barriers to the implementation of mathematical modeling in school curricula raised there from the perspective of the on going work of the models and modeling research group. In doing so we stress the need for critical literacy as well as the need to initiate a new research agenda based on the fact that we are now living in a fundamentally different world in which reality is characterized by complex systems. This may very well require us to go beyond conventional notions of modeling.

Surveying Theories and Philosophies of Mathematics Education

Any theory of thinking or teaching or learning rests on an underlying philosophy of knowledge. Mathematics education is situated at the nexus of two fields of inquiry, namely mathematics and education. However, numerous other disciplines interact with these two fields, which compound the complexity of developing theories that define mathematics education. We first address the issue of clarifying a philosophy of mathematics education before attempting to answer whether theories of mathematics education are constructible.