Difference between revisions of "CUMULATE parametrized asymptotic knowledge assessment"
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* the blue line denotes a successful use of a concept in a new problem (''pV'' = .5) | * the blue line denotes a successful use of a concept in a new problem (''pV'' = .5) | ||
* the green line denotes a penalty coefficient -- 1/''(<sub>succ</sub>att<sub>p</sub>''+2)''<sup>OPP</sup>'' -- as if it was the same problem (''OPP'' = .25) | * the green line denotes a penalty coefficient -- 1/''(<sub>succ</sub>att<sub>p</sub>''+2)''<sup>OPP</sup>'' -- as if it was the same problem (''OPP'' = .25) | ||
| − | * the red line ''merges'' the | + | * the red line ''merges'' the two graphs above |
| − | |||
[[Image:CUMULATE parameterized asymptotic knowledge assessment - knowledge growth and penalty.png]] | [[Image:CUMULATE parameterized asymptotic knowledge assessment - knowledge growth and penalty.png]] | ||
| − | <br/>([http://chart.apis.google.com/chart?cht=lxy&chs=300x240&chd=t:0,10,20,30,40,50,60,70,80,90,100 | + | <br/> |
| + | ([http://chart.apis.google.com/chart?cht=lxy&chs=300x240&chd=t:0,10,20,30,40,50,60,70,80,90,100%7C00,50,75,88,94,97,98,99,100,100,100%7C-1%7C-1,100,84,76,71,67,64,61,59,58,56%7C-1%7C00,50,63,66,66,65,63,61,59,58,56&chco=24588E,4C9B46,CF1E2B&chxt=x,y&chxl=1:%7C0%7C.2%7C.4%7C.6%7C.8%7C1%7C0:%7C0%7C1%7C2%7C3%7C4%7C5%7C6%7C7%7C8%7C9%7C10&chm=o,24588E,0,-1,10%7Co,4C9B46,1,-1,10%7Co,CF1E2B,2,-1,10&chg=10,20&chdl=knowledge+level%7Cpenalty+coefficient%7Cresulting+curve&chdlp=b+&chls=1,1,0%7C1,1,0%7C3,6,3&chtt=Knowledge+level+growth+and+change+in+penalty%7Ccoefficient+vs.+number+of+successful+attempts&chdlp=b alternatively] via [http://code.google.com/apis/chart/ Google Chart API]) | ||
==Example 2== | ==Example 2== | ||
| − | Below the speed of problem learning is held constant (''pV'' = .5) while the penalty for practicing the same problem changes. | + | Below: 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]] | ||
| − | <br/>([http://chart.apis.google.com/chart?cht=lxy&chs=300x240&chd=t:0,10,20,30,40,50,60,70,80,90,100 | + | <br/> |
| + | ([http://chart.apis.google.com/chart?cht=lxy&chs=300x240&chd=t:0,10,20,30,40,50,60,70,80,90,100%7C0,50,75,87,93,97,98,99,100,100,100%7C-1%7C0,47,71,83,90,94,97,98,98,99,100%7C-1%7C0,42,64,77,85,89,93,95,96,97,98%7C-1%7C0,35,54,66,73,79,83,86,88,90,92%7C-1%7C0,30,45,55,62,67,70,74,76,78,80%7C-1%7C0,25,38,45,51,55,58,61,63,65,66&chco=24588E,4C9B46,F3A030,CF1E2B,78387B,7C807F&chxt=x,y&chxl=1:%7C0%7C.2%7C.4%7C.6%7C.8%7C1%7C0:%7C0%7C1%7C2%7C3%7C4%7C5%7C6%7C7%7C8%7C9%7C10&chm=o,24588E,0,-1,10%7Co,4C9B46,1,-1,10%7Co,F3A030,2,-1,10%7Co,CF1E2B,3,-1,10%7Co,78387B,4,-1,10%7Co,7C807F,5,-1,10&chg=10,20&chdl=OPP=.01%7COPP=.10%7COPP=.25%7COPP=.50%7COPP=.75%7COPP=1.0&chdlp=b&chtt=Knowledge+level+growth+for+different+penalty%7Cparameters+vs.+number+of+successful+attempts alternatively] via [http://code.google.com/apis/chart/ Google Chart API]) | ||
==Example 3== | ==Example 3== | ||
| Line 41: | Line 42: | ||
=Studies= | =Studies= | ||
| − | * [[CUMULATE_user_and_domain_adaptive_user_modeling#Study_1|Study 1]] - comparison of the legacy and parametrized algorithms based on SQL problem-solving data | + | * [[CUMULATE_user_and_domain_adaptive_user_modeling#Study_1|Study 1]] - comparison of the [[CUMULATE_asymptotic_knowledge_assessment|legacy]] and [[CUMULATE_parametrized_asymptotic_knowledge_assessment|parametrized]] algorithms based on SQL problem-solving data |
= Contacts = | = Contacts = | ||
[[User:Myudelson | Michael V. Yudelson]] | [[User:Myudelson | Michael V. Yudelson]] | ||
Latest revision as of 21:50, 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
- succattp - 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/(succattp+2)OPP -- as if it was the same problem (OPP = .25)
- the red line merges the two graphs above
(alternatively via Google Chart API)
Example 2
Below: 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
