Herein we describe a mathematical model for the relation between in-use and discarded ceramic vessels. Following Schiffer (1976:53), we consider pre
historic and modern activities that create pathways for material objects to become deposited in the archaeological record. Accidents and use are pathways for breakage and damage to ceramic vessels. Broken or damaged vessels may be discarded and thereby incorporated into the archaeological record, or before being discarded they may be recycled for some other purpose. For the remainder of this paper, we consider a vessel to be discarded when it is removed from service as a cooking vessel; however, we recognize that there is reuse/recycling of pottery for other purposes (Deal and Hagstrum 1995). For example, a prominent use of broken vessels in the Mantaro Valley is as roof tiles or as protection for the top of an adobe wall. Another primary use for broken ceramic vessels is household storage. Deal and Hagstrum (1995) describe ceramic reuse for the Mantaro Valley Wanka, so we do not discuss it further here except to note that the vast majority (97 percent) of ceramic recycling by the Wanka simply delays deposition of some fraction of the broken ves sel into the archaeological record. We do recognize that there are forms of vessel recycling that remove ceramics from direct visibility in the archaeological record, such as grinding sherds for temper in ceramic vessel production. One pathway for ceramic vessel discard is accidental breakage unrelated to any intended use for food preparation. This may occur from accidental massive impact (Tani 1994:60), such as being dropped on the ground or being hit by a heavy object. Examples described by Mantaro informants include breakage due to natural hazards (wind blew the kitchen roof down), animals (chicken knocked it off table), children (goddaughter broke it playing with stick), and transport (broken when bringing it home on day of purchase). By definition, this form of breakage will be equally likely for all vessels, independent of their age or previous usage. Another pathway for ceramic discard is use in the performance of an activity, such as cooking. The use of a ceramic vessel creates the possibility that the vessel will be damaged, and the cumulative effect of even slight damage during use may eventually lead to complete failure. This form of breakage occurs preferentially in older vessels—those that have been in service and received repeated thermal shock (Bronitsky and Hamer 1986:96-97) or other damage, which are therefore more susceptible to breakage by weak impact (Tani 1994:60).
For breakage by either cause, the number of vessels discarded with a given age, ND (age), is related to the number of in-use vessels with that age, Nv (age), and the annual probability of breakage, PB, as follows:
which may be rearranged to solve for the total number of vessels in use:
The above expression allows a modern vessel-in-use age distribution to provide an estimate of vessel uselife. We use this expression to derive uselife for Wanka vessels from our contemporary survey of in-use vessel ages. Schiffer (1975:840) presented a relation between the total number of artifacts discarded, Td, the number of artifacts normally in use, S, the time period of use, t, and the artifact uselife, L
where age is the number of years of usage. The number of vessels that survive breakage to become a year older, Nv (age+1), is given by the difference between those in use and those discarded during the previous year:
These equations provide a relationship for the age distribution of vessels in use, Nv (age ), and those discarded, Nd (age), from which vessel uselife can be derived. Using the notation of Equation 1, vessel uselife is defined as the mean vessel age at the time of discard:
The age distribution of vessels in use can provide an estimate of uselife, L , since the number of dis-
carded vessels is just the difference in the number of vessels in use between successive years. By substituting Equation 2 into Equation 3, the uselife becomes:
To apply these equations to archaeological ceramic data, the numbers of vessels discarded during a given time period must be estimated, TD, perhaps from sherd collections for the purposes of this discussion (e.g., Hagstrum and Hildebrand 1990). In addition, the vessel uselife, Lmean, must be estimated. Hildebrand (1978) provides an alternative formulation of this problem in terms of object use or use number. To obtain human population estimates from the number of vessels in use, S, requires additional estimates of the number of vessels per household and household size (Foster 1960:606; Nelson 1991:170).
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