subtracting the intermediate steps. The students could keep track of the strokes of the intermediate steps with colored pencils in an elaborated system of chunking. A simplified multiplication table can be made by first finding the products of each base digit, then the products of the bases and the sub-bases, and finally the product of each sub-base: These tables are functionally complete for multiplication operations using Kaktovik numerals, but for factors with both bases and sub-bases it is necessary to first disassociate them: 6 * 3 = 18 is * = ( * ) + ( * ) = In the above example the factor (6) is not found in the table, but its components, (1) and (5), are. Legacy The Kaktovik numerals have gained wide use among Alaskan Iñupiat. They have been introduced into language-immersion programs and have helped revive base-20 counting, which had been falling into disuse among the Iñupiat due to the prevalence of the base-10 system in English-medium schools. When the Kaktovik middle school students who invented the system graduated to the high school in Barrow, Alaska (now renamed Utqiaġvik), in 1995, they took their invention with them. They were permitted to teach it to students at the local middle school, and the local community Iḷisaġvik College added an Inuit mathematics course to its catalog. In 1996, the Commission on Inuit History Language and Culture officially adopted the numerals, and in 1998 the Inuit Circumpolar Council in Canada recommended the development and use of the Kaktovik numerals in that country. Significance Scores on the California Achievement Test in mathematics for the Kaktovik middle school improved dramatically in 1997 compared to previous years. Before the introduction of the new numerals, the average score had been in the 20th percentile; after their introduction, scores rose to above the national average. It is theorized that being able to work in both base-10 and base-20 might have comparable advantages to those bilingual students have from engaging in two ways of thinking about the world. The development of an indigenous numeral system helps to demonstrate to Alaskan-native students that math is embedded in their culture and language rather than being imparted by western culture. This is a shift from a previously commonly held view that mathematics was merely a necessity to get into college/university. Non-native students can see a practical example of a different world view, a part of ethnomathematics. In