Bill Barton

Making sense of ethnomathematics: Ethnomathematics is making sense

There are unacknowledged difficulties within the literature of culture and mathematics, specifically in the use of the term ethnomathematics. As a step towards a more coherent approach, a framework to review the literature is proposed, and three authors examined. From this analysis, a definition of ethnomathematics is derived and elaborated. Two examples are reviewed as a test of the power of the definition and the resultant description of ethnomathematics.

A Comment on: Rowlands & Carson ``Where would formal, academic mathematics stand in a curriculum informed by ethnomathematics? A critical review''

The critique of ethnomathematics by Rowlands and Carson that appeared recently provides an opportunity to open debate on cultural issues in mathematics. This response argues that such debate must be based on contemporary writing in the field, and should not focus on extreme views within the political justification for ethnomathematics. It addresses some of the philosophical questions raised by Rowlands and Carson, and the relationship of the field with indigenous knowledge is raised. We also suggest that the role of ethnomathematics in mathematics education is now predominantly an empirical matter,and comment on some preliminary results from recent studies that indicate a positive role for culturally-based curricula.

Ethnomathematics and Philosophy

Any concept of ethnomathematics must eventually meet philosophical debates about the nature of mathematics. In particular neo-realist positions are anathema to the idea that mathematics is culturally based, but even modern quasi-empiricist philosophies are challenged by the fundamental relativity implied in ethnomathematical writing.

Teaching Mathematics as the Contextual Application of Mathematical Modes of Enquiry

Teachers whose lessons make a significant difference to students’ understanding of mathematical ideas appear to adapt mathematical modes to the restricted frames of school mathematics. We explored one of these frames, the preparation of teaching resources, to investigate our hypothesis about the central role of mathematical modes of enquiry. We set up an artificial resource preparation exercise amongst a group of knowledgeable mathematics educators and recorded their collaboration. We found that our personal mathematical modes were transforming, and the results of this process were embedded into our planning. We argue that teachers’ fluency with mathematical modes is the basis of their unique contribution in providing something that a textbook or annotated website cannot provide.

EAL Undergraduates Learning Mathematics.

This paper reports on a third study in a series aimed at investigating the issues surrounding the learning of mathematics at The University of Auckland by students who have English as an Additional Language (EAL students). As well as summarizing the earlier studies, it presents some new data showing that the problems experienced by EAL students are not experienced by first language English students (L1 students). The consequences of these studies for undergraduate mathematics at The University of Auckland are explored using English language proficiency data for the whole of the 2005 undergraduate mathematics cohort. This data shows that the consequences of having large numbers of EAL students in an undergraduate mathematics cohort are likely to be much more important than expected, and that recent initiatives to raise literacy entrance levels are having some effect.

Anthropological Perspectives on Mathematics and Mathematics Education

Viewed from a European knowledge perspective, much of the writing about mathematics and mathematics education might be regarded as instances of several different genres: including, for example, mathematical, philosophical, historical, sociological, educational, psychological, and a diverse corpus of writings concerned with mathematics and culture. This present chapter is specifically concerned with summarising and critiquing the literature pertaining to anthropological perspectives on mathematics and mathematics education,

Growing Understanding of Undergraduate Mathematics: A Good Frame Produces Better Tomatoes.

I lay a new theoretical framework across my own lecturing in order to understand what is happening. On the one hand, this is a test of the framework. On the other, I gain insights into both better practices and better course design. The framework constructs undergraduate teaching as the interaction between the discipline and the university pedagogical context. Overlaying this are three levels of teaching intent: pragmatic, epistemic, and heuristic. The resulting framework supports my growing understanding of lecturing practice. It also proves useful in analysing three characteristics of university mathematics: student responsibility for learning; enculturation of the discipline of mathematics; and the tyranny of examples. The framework is a tool for redesigning courses and developing delivery formats that are likely to enhance student learning and behaviour objectives of undergraduate mathematics. However, the analysis shows a deficiency of the framework in its paucity of attention to student learning in a university context. Extending the framework in this way is the next task.

The mathematical needs of secondary teachers: data from three countries

This article discusses the mathematical needs of secondary teachers, in particular senior secondary teachers who are preparing students for tertiary mathematical study. It also reports on data concerning secondary mathematics teacher qualifications in New Zealand, Australia and the United Kingdom. Despite an increasing need for both deeper and wider mathematical understanding, the levels of mathematical qualifications of secondary teachers in all three countries appear to have declined since the 1970s, with the current proportion of teachers without post-secondary mathematics qualifications being between 20 and 30%. The implications for mathematics learning are discussed and it is recommended that all second year mathematics teachers should have the equivalent of second year university mathematics.

Preface to “Ethnomathematics and Philosophy”

We who undertake ethnomathematical studies still have a philosophical problem. I still believe that a version of mathematical relativity is one of our basic assumptions. That is, the study of ethnomathematics rests on the idea that there can be more than one form of mathematics—it is neither absolute nor Platonist. Postmodern writing is exposing more about the contingent nature of mathematics, and the historicity of mathematics is becoming more accepted. Subjectivity and objectivity are becoming blurred (Brown 2011; Radford et al. 2008). But we are still a long way from wide agreement on a philosophical position that “allows” ethnomathematical investigations.

Mathematics, Education, and Culture: A Contemporary Moral Imperative

In 1984 Ubiratan D’Ambrosio gave a plenary address at ICME-5 in Adelaide that set a new direction for a major research effort in socio-cultural issues in mathematics education. His recent work uses the metaphor of mathematics as a “dorsal spine” on which monsters, not beautiful creatures, are often built. What must we do, what action must we take, to prevent ourselves from building monsters with mathematics and in mathematics education? This paper argues that theoretical approaches drawing on ecological concepts can lead us to understand the interconnectedness of teaching and scholarship with culture and society. I postulate three principles for action that may help guide moral behaviour within our discipline.