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The unit of analysis in mathematics education: bridging the political-technical divide?

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Abstract

Mathematics education is a complex, multi-disciplinary field of study which treats a wide range of diverse but interrelated areas. These include the nature of mathematics, the learning of mathematics, its teaching, and the social context surrounding both the discipline and applications of mathematics itself, as well as its teaching and learning. But research and researchers in mathematics education fall loosely into two camps: On the one hand there is technical research, drawing on mathematics, psychology and pedagogy, concerned with narrow questions about the teaching and learning of mathematics. On the other hand there is political and social research drawing on sociology and philosophy, addressing large scale problems of social consequence. These two camps tend to draw on different theoretical underpinnings as well as having different interests. Is it possible to find a shared theoretical element, a single unit of analysis for mathematics education which provides a unified approach to both analysing and explaining all of these diverse aspects? Can such a unit provide a bridge across the technical-political divide? Methodological and ontological senses of the term ‘unit of analysis’ are distinguished. Units of analysis in the ontological sense are proposed for each of the four listed subdomains of mathematics education. Drawing on Blunden’s (2009, 2010) interdisciplinary version of cultural historical activity theory a single over-arching unit of analysis, the collaborative project, is proposed for the whole field of mathematics education. This is applied to the case study of a single learner, illustrating the practical utility of this approach, as well as the way the technical-political divide might be bridged in our field.

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Notes

  1. Mathematics has a long tradition of calculation aids and technologies (abacus, Napier’s bones, logarithm tables, ready reckoners, slide rules, mechanical calculators, electronic calculators, computers) that have been influential in teaching mathematics. Mathematical apparatus such as the compasses, divider and ruler (or straight edge) have been in use since ancient times. For over a century specially designed mathematics teaching apparatus and manipulatives of all sorts have been developed, including Montessori’s golden beads, Cuisenaire rods, Dienes’ blocks, etc. One of the major mathematics teaching organisations in the United Kingdom, the Association of Teachers of Mathematics was originally called the Association for Teaching Aids in Mathematics when formed in 1955. The list of aids indicates the powerful impact of various technologies used in mathematics and its teaching, without mentioning symbolic technologies such as Hindu-Arabic numerals, decimal place value notation, algebraic notation, geometric figures, logical and proof notations, statistical tables and tests, computer software, etc. This illustrates the technological, tool-based roots of much of mathematical pedagogy.

  2. J. Evans (personal communication, 2011) argues that the methodological/ontological distinction is not as clear cut as I claim, and that what I have termed methodological uses of the term unit of analysis have deeper ontological implications than just statistical convenience. In some ways this parallels Quine’s (1969) argument that the range of the variables involved in any theory or language usage defines or enlarges the boundaries of its ontological commitments.

  3. The molecule is an irreducible unit—not because it does not have parts—but because further analysis into elements or particles results in the loss of its characteristics, its chemical properties.

  4. Núñez (2009), drawing on Engeström (2001), distinguishes between three generations of Activity Theory based on the work of: 1. Vygotsky, 2. Leontyev, and 3. Engeström, each with their own unit of analysis.

  5. A possible alternative representation is a tetrahedron connecting two Participants, Project and Mediating Artefact, to indicate the important roles of both Project and Mediating Artefact in the unit of analysis.

  6. A social critique of the concept of understanding is anticipated in the work of Mellin-Olsen (1981), the co-definer with Skemp (1976) of the relational-instrumental understanding dualism.

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This is a revised and extended version of the paper presented at the First Mathematics Education and Contemporary Theory conference, 17-19 July 2011, Manchester, UK.

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Ernest, P. The unit of analysis in mathematics education: bridging the political-technical divide?. Educ Stud Math 92, 37–58 (2016). https://doi.org/10.1007/s10649-016-9689-4

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