Revision as of 21:47, 20 January 2010

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

(alternatively) via Google Chart API)

To the left 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 two graphs above

Example 2

To the left: the speed of problem learning is held constant (pV = .5) while the penalty for practicing the same problem changes.

(alternatively via Google Chart API)

Example 3

Graph below shows differences the growth of knowledge while speed of learning varies. No penalty is given.

(alternatively via Google Chart API)

Studies

Study 1 - comparison of the legacy and parametrized algorithms based on SQL problem-solving data

Contacts

Michael V. Yudelson

@@ Line 28: / Line 28: @@
 ==Example 2==
-Below the speed of problem learning is held constant (''pV'' = .5) while the penalty for practicing the same problem changes.
+To the left: the speed of problem learning is held constant (''pV'' = .5) while the penalty for practicing the same problem changes.
-[[Image:CUMULATE parameterized asymptotic knowledge assessment - knowledge growth for diff OPP.png]]
+[[Image:CUMULATE parameterized asymptotic knowledge assessment - knowledge growth for diff OPP.png|thumb|300px|([http://bit.ly/4BaF6A alternatively] via [http://code.google.com/apis/chart/ Google Chart API])]]
-<br/>([http://chart.apis.google.com/chart?cht=lxy&chs=300x240&chd=t:0,10,20,30,40,50,60,70,80,90,100|0,50,75,87,93,97,98,99,100,100,100|-1|0,47,71,83,90,94,97,98,98,99,100|-1|0,42,64,77,85,89,93,95,96,97,98|-1|0,35,54,66,73,79,83,86,88,90,92|-1|0,30,45,55,62,67,70,74,76,78,80|-1|0,25,38,45,51,55,58,61,63,65,66&chco=24588E,4C9B46,F3A030,CF1E2B,78387B,7C807F&chxt=x,y&chxl=1:|0|.2|.4|.6|.8|1|0:|0|1|2|3|4|5|6|7|8|9|10&chm=o,24588E,0,-1,10|o,4C9B46,1,-1,10|o,F3A030,2,-1,10|o,CF1E2B,3,-1,10|o,78387B,4,-1,10|o,7C807F,5,-1,10&chg=10,20&chdl=OPP=.01|OPP=.10|OPP=.25|OPP=.50|OPP=.75|OPP=1.0&chdlp=b&chtt=Knowledge+level+growth+for+different+penalty|parameters+vs.+number+of+successful+attempts alternatively] via [http://code.google.com/apis/chart/ Google Chart API])
 ==Example 3==

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