I am developing a language learning program, and would like to know the mathematical expression and description for what I have (perhaps incorrectly) calling a recursive series method of learning the order of a foreign alphabet.
In sum, the learner would write out the new alphabet (I'll use English as an example) in order, one line at a time:
Example:
a ab abc abcd {...} abcdefghijklmnopqrstuvwx abcdefghijklmnopqrstuvwxy abcdefghijklmnopqrstuvwxyz
In initial tests, this has shown to be a highly time-efficient way of getting illiterate students to accurately recall the name, sound, and order of an alphabet.
As I consider further research and even publication, I would like to know what to call this pattern, in mathematical language.
If it helps, the pattern is basically
From 1 to 10
1 1,2 1,2,3 1,2,3,4 1,2,3,4,5 1,2,3,4,5,6 1,2,3,4,5,6,7 1,2,3,4,5,6,7,8 1,2,3,4,5,6,7,8,9 1,2,3,4,5,6,7,8,9,10
Notice that each "column" of numbers is written multiple times (except for the final number, of course) and the ascending order of the series is repetitive enough that a reader/student will learn the order of whatever ordered set you throw at him in a short amount of time.
Basically: is there a way of using algorithmic notation to say: given a full set (in this case, the alphabet), iterate through each element along with its preceding elements, until you have displayed the whole set?
I keep thinking of it being represented as something like f(n) = f(n-1) + f(n-2)... but I don't want to just guess at it.
I appreciate your help in describing these patterns algorithmically, and in terms of what kind of recursion is going on there.