Review: Saxe, Cultural development of mathematical ideas

Saxe, Geoffrey B. 2012. Cultural development of mathematical ideas.  Cambridge: Cambridge University Press. 393 pp.

Reviewed by Summar Saad (Wayne State University)

In Cultural Development of Mathematical Ideas, Geoffrey B. Saxe takes an ambitious approach in exploring the cultural and cognitive origins of mathematical thought. Using an extensive number of experiments oriented towards the particular practices of the Oksapmin of Papua New Guinea, Saxe demonstrates that individual action in relation to collective activities such as economic exchange and schooling is the “locus of both the reproduction and the alteration of cultural phenomena, whether collective practices of daily life or cultural forms of representation” (p.191). His conceptual framework, which follows Sperber and Hirschfeld’s critique of the conceptualization of culture as fixed and bounded entities rather than a “property that representations, practices, and artifacts possess to the extent that they are caused by population-wide distribution processes (p.17),” seeks to illustrate the heterogeneity and permeability of cultural and cognitive processes through activity using synchronic and diachronic perspectives.

In Part I, Saxe begins by framing the scope of his study using relevant scholarly work on the area of cognition and mathematical thought, and details his time in the field in 1978, 1980, and 2001. His approach interrogates three genetic processes: 1) microgenesis, the transformation of the body-part form into a vehicle to represent values numerically, 2) sociogenesis, which involves the microgenetic activities of multiple individuals at multiple sites that, collectively, constitute a process in which representational forms and functions are reproduced and altered in a community over time, and 3) ontogenesis, which involves the shifts in form-function relations in activity over the course of an individual’s development (p.29). Using a number of helpful figures, Saxe also unravels the specifics of the Oksapmin body-counting system where one begins counting with the thumb on one hand of the body and continues across to the opposite side ending at the little finger. Familiarity with this system is important for his investigation into how this system changes through time. By this point, the readers are ready to enter the field, where Saxe attempts to tease out the historical and social processes at play in the way the Oksapmin respond to mathematical challenges in collective activities under shifting conditions.

In Part II, Saxe traces the history of Papua New Guinea from a pre-contact period where people traded commodities including a shell currency (bonang) and through the sustained contact of the Oksapmin communities with Western societies, which led to the proliferation of trade stores that supported cash as the universal medium for exchange. Focusing on the activity of economic exchange, Saxe asserts that “with increasing participation in the money economy associated with Oksapmin cohorts, we find a shift from external correspondences that serve numerical functions to internal correspondences that serve arithmetical functions” (p.95). Saxe then shifts his focus from knowledge of Oksapmin body-part counting and Tok Pisin representational forms to look at the semiotic forms people use to represent the objects of economic exchange. What he discovers is that over time, as people began using the body system to quantify currency, reciprocally the currency system became incorporated into the structure of the body-counting system. Ultimately Saxe demonstrates that cognitive processes exhibit uniformity and variation in a single period as well as unity and discontinuity over historical time.

In Part III, Saxe reviews the transformation of schooling in Oksapmin beginning in the early 1960s and following the introduction of “Western schooling,” Bible school, and community schooling. Rather than conceptualizing schooling as a direct cause of cognitive development, Saxe focuses on the dynamics of the reproduction and alteration of the forms of numerical representation and the functions they serve as students and teachers participate in collective practices of classroom life (p. 194). He establishes the ways in which Oksapmin children reproduce the body form as they solve arithmetical problems in the same way that they produce variants in the body form, “inadvertently altering the use of the system to serve new functions (p.236).” With recent educational reforms, he notes a shift in the teaching of mathematics using only English to using Tok Pisin and Oksapmin as well. Saxe observes that with this shift came others including the use of stones in classroom computations as well as a developing facility with Hindu-Arabic-based algorithms.

In Part IV, Saxe brings the discussion full circle and returns to the implications of his findings with regards to his conceptual framework. In his analysis, he points to three key properties that emerge in form-function relations: conventionality, hybridity, and instrumentality. Using parallels from evolutionary biology, he also takes up the question of how form-function relations develop and why. Though not an anthropologist, Saxe deserves great credit for his ethnographic treatment of this serious cognitive question. Saxe successfully presents an alternative methodological approach to our understandings of culture and cognition that does not treat them as independent but rather as an interplay of the two. His work offers great insight into cognition and culture as processes rooted in a multiplicity of contexts and activities. Although this book covers a lot of ground, and is sometimes very abstract, the organization and flow of the content is seamless and easy to follow. Saxe also takes great care to account for any threats to validity, and while each of the eighteen individual studies he conducts are not without their flaws; the overall picture shines clearly at the end: culture and cognition are processes interwoven and linked through activity.



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