CUMULATE's parameterized asymptotic knowledge assessment algorithm is an attempt to overcome shortcomings of its non-paramterized version. Namely, one-fits-all nature, prohibiting parameter tuning for individual users' abilities and problem complexities.

Computation

The formula below is used to update the knowledge levels of concepts (c) addressed in a problem (p). This formula reflects the following principles (identical to the predecessor algorithm).

there are several domain concepts (knowledge items, rules, productions) involved in solving a problem; the knowledge of each of them is updated proportionally to the others
knowledge is updated only upon correct user answers, there is no penalty for errors
solving a problem correctly multiple times will result in diminishing update (growth) of the knowledge level of the concepts as the number of successes grows

in addition:

the initial level of knowledge, speed of knowledge growth, and penalty for repetitive (correct) solutions to the problem - are now adjustable parameters

, where

Ko - is the starting level of knowledge, Ko ∈ [0, 1]
res - result of user action (0 -error, 1 - correct);
Wc,p - is a weight of concept c in problem p
ΣWc,p - is the sum of weights of all concepts in problem p
_succatt_p - is a number of successful solutions to problem p prior to current attempt
pV - speed of knowledge growth parameter, pV ∈ [0, 1]
OPP - over-practicing parameter, controlling the penalty for repetitively solving one problem (correctly), OPP ∈ [0, 1]

Examples

Example 1

Below is a graph of concept's knowledge level growth and the penalty coefficient vs. number of successful attempts to apply it in a problem. The lines denote:

the blue line denotes a successful use of a concept in a new problem (pV = .5)
the green line denotes a penalty coefficient -- 1/(_succatt_p+2)^OPP -- as if it was the same problem (OPP = .25)
the red line merges the prior two graphs

(alternatively via Google Chart API)