Elementos de Análisis en Matemáticas Quichua y Castellano (Elements of Analysis in Quichua and Spanish) by Consuelo Yánez CossÍo and AugustÍn Jerez of the Pontificia Universidad Católica in Quito, Ecuador.

The authors start from the premise that different languages not only have differences in basic mathematical vocabulary, but also in morphological structure, operational mechanisms and situations in which mathematics is used. In societies without written communications calculations are made to handle various social and commercial situations, but the development of those mathematical systems is often retarded by the imposition of the system used by the socially and politically dominant culture.

In traditional Quichua culture in Ecuador children learn to handle mathematical concepts and operations at an early age as a natural part of their participation in the system of production. Currently not everyone who began the traditional learning process has finished it. Some received interference from mathematics of the Spanish language culture, some have participated in activities that do not require knowledge of traditional Quichua mathematics.

One important difference between Quichua and Spanish mathematics is that learning of Quichua mathematics is always bound to concrete social applications while the learning of Spanish mathematics is very often devoid of such real life applications.

Another important difference is that the number names in Quichua relate clearly to the base ten numeration system. For example the Quichua expression for 222 is ishcai patsac ishcai chunca ishcai which can be translated directly as two hundred two ten two. The Spanish expression for 222, doscientos veintidos, does not relate so directly to its meaning in base ten. This lack of direct and obvious relation with the structure of base ten may be partly responsible for memoristic rather than meaningful learning.

Quichua culture has a space-time system that is spiral rather than linear. The authors hypothesize that their mathematical systems are also spiral in nature. They see years as cycles related to the crop cycle rather than a linear array of 365 days. Another manifestation of the spiral/circular nature of their mathematical system is a device made of bone called a "huari" that is analogous to a die, but is more circular than cubical.

Operations within the Quichuan mathematical system are made at three levels: the first level is concrete and called "graneo", on a second level words are used to express quantities and at the third level there are mental calculations without overt words or symbols. The fundamental arithmetical processes tend to be based on tens and fives. For instance to add 266 and 288 the process can be related more or less as follows:

266 = 200 + 50 + 10 + 5 + 1

+288 = 200 + 50 + 30 + 5 + 3

400 + 100 = 500 (picha patsac)

40 + 10 = 50 (picha chunca)

3 + 1 = 4 (chusca)

554

(picha patsac picha chunca chusca)

The term "graneo" refers to the use of grains of corn, beans, seeds, pebbles, etc. either directly or with an abacus-like device called a "Contador del Cañar" (Cañar Counter).

Mental calculations can be quite sophisticated. One informant expressed the proces of mental calculation as follows:

In the mind there's a kind of path with spaces for numbers; there's a space divided in 10, then spaces that indicate each of the tens until 90, then the hundreds, the thousands and like that all the rest. Everyone has their path and there they mark the quantities they need to add and subtract. Each ten has its units and each hundred its tens.

The "paths" that the informant indicated tend to be circular with "nodes" that indicate the tens, hundreds, thousands, etc.

Work still needs to be done to translate what is know about Quichua mathematics into the school curriculum so that it can help facilitate learning of mathematics among Quichua children.