Abstract
Histories of mathematical practice account for mathematical knowledge and action by interpreting presently-available evidence as traces of the events, contexts, and relations that make up the past. Interpretations depend on the assumptions one makes about how mathematical knowledge works, insofar as it is knowledge and insofar as it is mathematical. Though the specific rules and their meanings can differ from context to context, mathematics is a kind of ruly knowledge, expected to follow orderly patterns and principles wherever it is found. The contexts and activities of mathematical practice—how that knowledge is made, shared, applied, and understood—are necessarily less ruly, and different practices leave or occlude different kinds of evidence for historical interpretation. The apparent ruliness of mathematics can be both a resource and an obstacle for understanding its unruly pasts. Historians’ interpretive assumptions and goals have been shaped by centuries of interaction between mathematics research, history, and education.As a guide for mathematics educators and education researchers to historical perspectives on mathematical practice, this article briefly introduces four major interpretive traditions that inform the present discipline of mathematics history. It then illustrates some interpretive approaches and challenges through the history of blackboards in mathematical practice before explaining the two broad kinds of historical interpretation applied to mathematical practices. Reconstruction involves understanding the conditions and contexts of practices in a single historical moment. Genealogy, by contrast, connects elements of the past across time through transmission, interpretation, adaptation, and other kinds of preservation and change.
Highlights
Not every piece of mathematical writing intended to last in perpetuity survives for historical interpretation, and all are altered through transmission, conservation, and use
Different methods, and different assumptions support different ways of understanding how mathematical ideas and practices relate to each other and to their wider contexts, how they change, and how they become manifest in texts and other artifacts
Traces gain meaning through a multiplication of contexts: the 1806 École Polytechnique note about repainting blackboards gives information about mathematical practice because, from other sources, we know something about the role of mathematics in the school’s curriculum, the conceptual elements of that curriculum that survive in other pedagogical media such as posters or textbooks, the relationship between administrative records and the actions they document and authorize, the material parameters of writing with chalk on painted surfaces, the living traditions of lecturing and examination in mathematics and other fields, and much else
Summary
Numbers mostly add up, circumferences and diameters have about the same ratios, ruler and compass constructions will not ordinarily trisect an angle, no matter the surrounding culture or context This ruliness makes it possible to infer quite a lot from limited, indirect evidence of past practices: if a practitioner records (or even merely implies) just the inputs and outputs of a calculation, in many circumstances one can confidently guess at the calculation’s omitted parts. Putting the past in context means reconstructing relationships within a single moment and tracing genealogies of change over time These two aims demand reckoning with the unruly pasts of ruly knowledge
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