Roman numerals is a well-known example of a number system that does not just count up. This is a form of subtractive notation [wiki] which transformed Roman numerals from a variation of tallying to a positional number system, where the order of the digits dictates the value of the number.
Roman numerals use letters to represent numbers:
For example, the number 158 is written as CLVIII (100+50+5+1+1+1). Numerals are generally written in decreasing order. However, if a smaller numeral is placed before a larger one, then the value of the smaller is taken away from the larger. For example, 4 is written as IV (5-1) and 140 is written as CXL (100+(50-10)). It is likely that subtractive notation is used to avoid numbers becoming too long.
Roman numerals is not the only number system to use this method however, the Yoruba (Ethnic group mainly from Benin, Nigeria, and Togo) counting system also uses a similar method, which comes from a way of counting cowrie shells.
Yoruba Counting System
Cowrie shells were used all across West Africa as currency, which meant efficient ways of counting large quantities of shells was vital, shells were traded in standard quantities . This lead to the development of counting methods described by Adolphus Mann in 1887:
"When a bagful is cast on the floor, the counting person sits or kneels down beside it, takes 5 and 5 cowries, and counts silently, 1, 2, up to 20, thus 100 are counted off, this is repeated to get a second 100, these little heaps each of 100 cowries are united, and a next 200 is, when counted, swept together with the first. Such sums, as originate from counting cowries, are a sort of standard money, 20, 100 and then especially 200, and 400 is 4 little heaps each of 100 cowries, or 2 each of 200 cowrie" [Mann, 1887]
Numbers are thought of with respect to these standard quantities, counting up until it is more intuitive to think of the number as counting down from a larger quantity. For example, numbers are counted up with compound words until 14 ('merinlaa' translated as 'one for than 10') where the convention changes to counting down from 20, so 15 is expressed as 'meeedogun' which can be translated as 'twenty less 5'.
An example of how a large number such as 525 may have been counted is shown below, first start with 600 cowrie shells, counted by 3 lots of 200. 80 shells are then removed by counting out 4 lots of 20, before adding back in one group of 5. This is more efficient than being limited to solely counting up. [Huylebrouck, 2006]
What is perhaps the most interesting part of this system is the effect it has on manipulation and arithmetical skills, 'Yoruba conservation [of number] results are as good or better than those on American children [tested in the same way]' [Lloyd, 1981]. Finding the word for a number requires a significant amount of mental arithmetic, which is perhaps why the Yoruba counting system has been praised my many, as far back as 1887:
"We light, as it were, on a building, which, when viewed from base to summit is not behind our European systems in regularity and symmetry, while the system surpasses them in the aptitude of interlinking the separate members; it stands to them in the same relation as the profusely ornamented Moorish style stands to the more sober Byzantine" [Mann, 1887]