University of Haifa
Tel Aviv University
Early development of numerical thinking - the discursive view.
The point of departure for this talk is that the language in which researchers conduct their investigations influences their ability to ask questions and interpret data. The traditional language of research on numerical thinking implies that the child is aware of the existence of the abstract objects called numbers prior to being able to apply them in any way. Those who adopt discursive approach conceptualize thinking at large and numerical thinking in particular as forms of communication. In this way, they remove the assumption about the pre-existence of numbers: by portraying them as discursive constructs, they imply that numbers are products rather than pre-given objects of human communication. In this talk, after presenting the basic tenets of the discursive approach to cognition, we will explore the question of how the proposed reconceptualization impacts our understanding of numerical thinking and informs the practice of fostering children’s numerical development.Theoretical arguments will be supported with empirical examples coming from our own and other researchers’ recent studies.
Charles University in Prague
Language in Change, or how we changed the language of mathematics and how the language of mathematics changed us.
If we compare mathematical texts from the past with our contemporary mathematical practice, we will notice several differences. In the course of the history we witness a gradual increase of logical power (we can prove stronger theorems), increase of expressive power (we can study more complex phenomena), increase of methodic power (we have stronger analytic methods), increase of integrative power (our theories display deeper unity), increase of explanatory power (we can explain subtler phenomena), and increase of metaphorical power (we can apply the language to unforeseen situations) of the language of mathematics. We will illustrate the increase of these parameters on the transition from synthetic geometry of the ancient Greeks to algebra of the Arabic and early modern period. For each of the six aspects mentioned above we will describe the linguistic innovation by means of which it is formed. In teaching mathematics we teach our children, besides the many concepts, theorems, and proofs, also the linguistic innovations, which make possible to define these concepts, to formulate these theorems, and to mate these proofs. Sensitivity to language in which mathematics is developed and communicated is an important quality, which a good teacher has to cultivate.
University of London
Beyond communication: using language for researching curriculum, pedagogy and policy in mathematics education
The language and other media used in educational practices do not simply transmit the ideas and intentions of speakers and writers. The practices themselves shape and are shaped by what is said. The texts produced by speakers and writers in a practice such as mathematics education thus draw on ways of construing the world that are legitimate within the practice and also constrain the ways in which listeners and readers may respond. Discourse analytic methods drawing on Halliday’s systemic functional linguistics provide means of describing the world of mathematics education through analysis of its spoken and written texts. This allows us to address questions about the nature of mathematics and mathematical activity and about the nature of teaching and learning, of teachers and of students as these are produced in the texts of a given practice. In this plenary I will discuss a discourse analytic theoretical and methodological approach to researching mathematics education and will illustrate this with examples drawn from investigations of curriculum, policy and classroom practice.
APS – National Center for School Improvement
Images of Numeracy – the power of images in mathematical communication
The last decade we have seen a strong focus on usable mathematics: mathematics strongly connected to the real world. Related concepts to this approach are functional mathematics or numeracy. Such concepts can be arranged along a continuum of increasing levels of sophistication from formative to integrative. In the integrative phase numeracy is viewed as a complex multifaceted and sophisticated construct, incorporating the mathematics, communications, cultural, social, emotional and personal aspects of each individual in context. A closer look however at learning or test materials used in many different countries reveals that most materials merely consist of word problems or of exercises with formal arithmetic skills. One could say that the sophistication of the concepts runs way ahead of the sophistication of the learning and testing materials. In this era of technology and multimedia a next step can and should be made to bring real quantitative problems – problems as individuals face them - into learning or test materials by using real life images. In the Dutch programme for evidence based educational research (Onderbewijs) a grant was awarded to research the use of real life images in numeracy test materials. The lecture will focus on the development of the materials, on the results of the research to compare students’ performance on word problem with their performance on equivalent problem with image rich contexts, and on the effect it has on policy.