Learning from Learning
Curves using Learning Factors
Analysis
Hao Cen, Kenneth Koedinger, Brian Junker
Human

Computer Interaction
Institute
Carnegie Mellon University
Cen, H., Koedinger, K., Junker, B.
Learning Factors
Analysis

A General Method for Cognitive Model
Evaluation and Improvement
.
the 8th International
Conference on Intelligent Tutoring Systems
. 2006.
Science,
26(2)
Cen, H., Koedinger, K., Junker, B.
Is Over Practice
Necessary? Improving Learning Efficiency with the
Cognitive Tutor
.
The 13th International Conference on
Artificial Intelligence in Education (AIED 2007)
. 2007.
Student Performance As They
Practice with the LISP Tutor
Production Rule Analysis
Evidence for Production Rule as an
appropriate
unit of knowledge acquisition
Using learning curves to
evaluate a cognitive model
Lisp Tutor Model
Learning curves used to validate cognitive model
Fit better when organized by knowledge components
(productions) rather than surface forms (programming language
terms)
But, curves not smooth for some production rules
“Blips” in leaning curves indicate the knowledge
representation may not be right
Corbett, Anderson, O’Brien (1995)
Let me illustrate …
Curve for “Declare
Parameter” production rule
How are steps with blips different from others?
What’s the unique feature or
factor
explaining these
blips?
What’s happening
on the 6th & 10th
opportunities?
Can modify cognitive model using unique
factor
present at “blips”
Blips occur when to

be

written program has 2 parameters
Split
Declare

Parameter by parameter

number factor:
Declare

first

parameter
Declare

second

parameter
Learning curve analysis by hand
& eye …
Steps in programming problems where the function
(“method”) has two parameters (Corbett, Anderson,
O’Brien, 1995)
Can learning curve analysis be
automated?
Learning curve analysis
Identify blips by hand & eye
Manually create a new model
Qualitative judgment
Need to automatically:
Identify blips by system
Propose alternative cognitive models
Evaluate each model quantitatively
Overview
Learning Factors Analysis algorithm
A Geometry Cognitive Model and Log Data
Experiments and Results
Learning Factors Analysis (LFA):
A Tool for KC Analysis
LFA is a method for discovering & evaluating alternative
cognitive models
Finds knowledge component decomposition that best predicts
student performance & learning transfer
Inputs
Data: Student success on tasks in domain over time
Codes: Hypothesized factors that drive task difficulty
A mapping between these factors & domain tasks
Outputs
A rank ordering of most predictive cognitive models
For each model, a measure of its generalizability & parameter
estimates for knowledge component difficulty, learning rates, &
student proficiency
Learning Factors Analysis (LFA) draws
from multiple disciplines
Machine Learning & AI
Combinatorial search
(Russell & Norvig, 2003)
Exponential

family principal component analysis
(Gordon,
2002)
Psychometrics & Statistics
Q Matrix & Rule Space
(Tatsuoka 1983, Barnes 2005)
Item response learning model
(Draney, et al., 1995)
Item response assessment models
(DiBello, et al., 1995;
Embretson, 1997; von Davier, 2005)
Cognitive Psychology
Learning curve analysis
(Corbett, et al 1995)
Steps in Learning Factors Analysis
We’ve talked
about some of
these steps 1

4
before …
LFA
–
1. The Q Matrix
How to represent relationship between knowledge components
and student tasks?
Tasks also called items, questions, problems, or steps (in problems)
Q

Matrix (Tatsuoka, 1983)
2* 8 is a single

KC item
2*8
–
3 is a conjunctive

KC item, involves two KCs
13
Item  KC
Add
Sub
Mul
Div
2*8
0
0
1
0
2*8

3
0
1
1
0
What good is a Q matrix? Used to predict
student accuracy
on items not previously
seen
, based on KCs involved
LFA
–
2. The Statistical Model
Problem: How to predict student responses from model?
Solutions: Additive Factor Model (Draney, et al. 1995, Cen, Koedinger,
Junker, 2006)
LFA
–
2. An alternative “conjunctive”
model
Conjunctive Factor Model (Cen, Koedinger, Junker, 2008)
16
LFA

4. Model Evaluation
•
How to compare cognitive models?
•
A good model minimizes prediction risk by balancing fit
with data & complexity (Wasserman 2005)
•
Compare BIC for the cognitive models
•
BIC is “Bayesian Information Criteria”
•
BIC =

2*log

likelihood + numPar * log(numOb)
•
Better (lower) BIC == better predict data that haven’t seen
•
Mimics cross validation, but is faster to compute
LFA
–
5. Expert Labeling & P

Matrix
Problem: How to find the potentials to improve the
existing cognitive model?
Solution: Have experts look for
difficulty factors
that are
candidates for new KCs. Put these in P matrix.
Item  Skill
Add
Sub
Mul
2*8
0
0
1
2*8
–
3
0
1
1
2*8

30
0
1
1
3+2*8
1
0
1
Q Matrix
P Matrix
Item  Skill
Deal with
negative
Order
of Ops
…
2*8
0
0
2*8
–
3
0
0
2*8

30
1
0
3+2*8
0
1
LFA
–
5. Expert Labeling and P

Matrix
Operators on Q and P
Q + P[,1]
Q[, 2] * P[,1]
Item  Skill
Add
Sub
Mul
Div
neg
2*8
0
0
1
0
0
2*8
–
3
0
1
1
0
0
2*8

30
0
1
1
0
1
Q

Matrix after add P[, 1]
Item  Skill
Add
Sub
Mul
Div
Sub

neg
2*8
0
0
1
0
0
2*8
–
3
0
1
1
0
0
2*8

30
0
0
1
0
1
Q

Matrix after splitting P[, 1], Q[,2]
LFA
–
6. Model Search
Problem: How to find best model given P

matrix?
Solution: Combinatorial search
A best

first search algorithm (
Russell & Norvig
2002)
Guided by a heuristic, such as BIC
Start from an existing model
Combinatorial Search
Goal:
Do model selection within the logistic regression model
space
Steps:
1.
Start from an initial “node” in search graph
2.
Iteratively create new child nodes by splitting a model using
covariates or “factors”
3.
Employ a heuristic (e.g. fit to learning curve) to rank each
node
4.
Expand from a new node in the heuristic order by going back
to step 2
LFA
–
6. Model Search
Automates the process of
hypothesizing alternative KC
models & testing them against
data
Overview
Learning Factors Analysis algorithm
A Geometry Cognitive Model and Log Data
Experiments and Results
Domain of current study
15 skills
1.
Circle

area
2.
Circle

circumference
3.
Circle

diameter
4.
Circle

radius
5.
Compose

by

addition
6.
Compose

by

multiplication
7.
Parallelogram

area
8.
Parallelogram

side
9.
Pentagon

area
10.
Pentagon

side
11.
Trapezoid

area
12.
Trapezoid

base
13.
Trapezoid

height
14.
Triangle

area
15.
Triangle

side
Domain of study: the area unit of the geometry tutor
Cognitive model:
Log Data

Skills in the Base
Model
Student
Step
Skill
Opportunity
A
p
1
s
1
Circle

area
1
A
p
2
s
1
Circle

area
2
A
p
2
s
2
Rectangle

area
1
A
p
2
s
3
Compose

by

addition
1
A
p
3
s
1
Circle

area
3
The Split
Binary Split

splits a skill a skill with a factor
value, & a skill without the factor value.
Student
Step
Skill
Opportunity
A
p
1
s
1
Circle

area

alone
1
A
p
2
s
1
Circlearea

embed
1
A
p
2
s
2
Rectangle

area
1
A
p
2
s
3
Compose

by

addition
1
A
p
3
s
1
Circle

area

alone
2
Student
Step
Skill
Opportunity
Factor

Embed
A
p
1
s
1
Circle

area
1
alone
A
p
2
s
1
Circle

area
2
embed
A
p
2
s
2
Rectangle

area
1
A
p
2
s
3
Compose

by

addition
1
A
p
3
s
1
Circle

area
3
alone
After Splitting Circle

area by
Embed
The Heuristics
Good model captures sufficient variation in
data but is not overly complicated
balance between model fit & complexity minimizing
prediction risk (Wasserman 2005)
AIC and BIC used as heuristics in the search
two estimators for prediction risk
balance between fit & parisimony
select models that fit well without being too complex
AIC =

2*log

likelihood + 2*number of parameters
BIC =

2*log

likelihood + number of parameters *
number of observations
System: Best

first Search
an informed graph search algorithm
guided by a heuristic
Heurisitcs
–
AIC, BIC
Start from an existing model
System: Best

first Search
an informed graph search algorithm
guided by a heuristic
Heurisitcs
–
AIC, BIC
Start from an existing model
System: Best

first Search
an informed graph search algorithm
guided by a heuristic
Heurisitcs
–
AIC, BIC
Start from an existing model
System: Best

first Search
an informed graph search algorithm
guided by a heuristic
Heurisitcs
–
AIC, BIC
Start from an existing model
System: Best

first Search
an informed graph search algorithm
guided by a heuristic
Heurisitcs
–
AIC, BIC
Start from an existing model
System: Best

first Search
an informed graph search algorithm
guided by a heuristic
Heurisitcs
–
AIC, BIC
Start from an existing model
Overview
Learning Factors Analysis algorithm
A Geometry Cognitive Model and Log Data
Experiments and Results
Experiment 1
Q: How can we describe learning behavior in
terms of an existing cognitive model?
A: Fit logistic regression model in equation
above (slide 27) & get coefficients
Experiment 1
Results:
Skill
Intercep
t
Slope
Avg Opportunties
Initial Probability
Avg Probability
Final
Probability
Parallelogram

area
2.14

0.01
14.9
0.95
0.94
0.93
Pentagon

area

2.16
0.45
4.3
0.2
0.63
0.84
Student
Intercep
t
student
0
1.18
student
1
0.82
student
2
0.21
Model
Statistics
AIC
3,950
BIC
4,285
MAD
0.083
Higher intercept of skill

> easier skill
Higher slope of skill

> faster students learn it
Higher intercept
of student

>
student initially
knew more
The AIC, BIC & MAD
statistics provide
alternative ways to
evaluate models
MAD = Mean Absolute
Deviation
Experiment 2
Q: How can we improve a cognitive model?
A:
Run
LFA on data including factors &
search through model space
Experiment 2
–
Results with BIC
Splitting Compose

by

multiplication into two skills
–
CMarea and CMsegment, making a distinction
of the geometric quantity being multiplied
Model 1
Model 2
Model 3
Number of Splits:3
Number of Splits:3
Number of Splits:2
1.
Binary split compose

by

multiplication by
figurepart segment
2.
Binary split circle

radius by repeat repeat
3.
Binary split compose

by

addition by
backward backward
1.
Binary split compose

by

multiplication by figurepart
segment
2.
Binary split circle

radius by
repeat repeat
3.
Binary split compose

by

addition by figurepart area

difference
1.
Binary split compose

by

multiplication by
figurepart segment
2.
Binary split circle

radius
by repeat repeat
Number of Skills: 18
Number of Skills: 18
Number of Skills: 17
AIC: 3,888.67
BIC: 4,248.86
MAD: 0.071
AIC: 3,888.67
BIC: 4,248.86
MAD: 0.071
AIC: 3,897.20
BIC: 4,251.07
MAD: 0.075
Experiment 3
Q: Will some skills be better merged than if
they are separate skills? Can LFA recover
some elements of original model if we search
from a merged model, given difficulty factors?
A:
Run
LFA on the data of a
merged
model,
and search through the model space
Experiment 3
–
Merged Model
Merge some skills in the original model to remove some
distinctions, add as a difficulty factors to consider
The merged model has 8 skills:
Circle

area, Circle

radius => Circle
Circle

circumference, Circle

diameter => Circle

CD
Parallelogram

area and Parallelogram

side => Parallelogram
Pentagon

area, Pentagon

side => Pentagon
Trapezoid

area, Trapezoid

base, Trapezoid

height => Trapezoid
Triangle

area, Triangle

side => Triangle
Compose

by

addition
Compose

by

multiplication
Add difficulty factor “direction”: forward vs. backward
Experiment 3
–
Results
Model 1
Model 2
Model 3
Number of Splits: 4
Number of Splits: 3
Number of Splits: 4
Number of skills: 12
Number of skills: 11
Number of skills: 12
Circle *area
Circle *radius*initial
Circle *radius*repeat
Compose

by

addition
Compose

by

addition*area

difference
Compose

by

multiplication*area

combination
Compose

by

multiplication*segment
All skills are the same as those in
model 1 except that
1. Circle is split into
Circle
*backward*initial, Circle
*backward*repeat, Circle*forward,
2.
Compose

by

addition is not split
All skills are the same as those in
model 1 except that
1. Circle is split into
Circle
*backward*initial, Circle
*backward*repeat, Circle
*forward,
2.
Compose

by

addition is split
into Compose

by

addition and
Compose

by

addition*segment
AIC: 3,884.95
AIC: 3,893.477
AIC: 3,887.42
BIC: 4,169.315
BIC: 4,171.523
BIC: 4,171.786
MAD: 0.075
MAD: 0.079
MAD: 0.077
Experiment 3
–
Results
Recovered three skills (Circle,
Parallelogram, Triangle)
=>
distinctions made in the original model are necessary
Partially recovered two skills (Triangle, Trapezoid
)
=>
some original distinctions necessary, some are not
Did not recover one skill (
Circle

CD)
=>
original distinction may not be necessary
Recovered one skill (Pentagon
) in a different way
=>
Original distinction may not be as significant as
distinction caused by another factor
Beyond Experiments 1

3
Q: Can we use LFA to improve tutor
curriculum by identifying over

taught or
under

taught rules?
Thus adjust their contribution to curriculum length
without compromising student performance
A: Combine results from experiments 1

3
Beyond Experiments 1

3

Results
Parallelogram

side is over taught.
high intercept (2.06), low slope (

.01).
initial success probability .94, average number of practices per student is
15
Trapezoid

height is under taught.
low intercept (

1.55), positive slope (.27).
final success probability is .69, far away from the level of mastery, the
average number of practices per student is 4.
Suggestions for curriculum improvement
Reducing the amount of practice for Parallelogram

side should save
student time without compromising their performance.
More practice on Trapezoid

height is needed for students to reach
mastery.
Beyond Experiments 1

3

Results
How about Compose

by

multiplication?
Intercept
slope
Avg
Practice
Opportunties
Initial
Probability
Avg
Probability
Final
Probability
CM

.15
.1
10.2
.65
.84
.92
With final probability .92 students seem to have mastered
Compose

by

multiplication.
Beyond
Experiments 1

3

Results
However, after split
CMarea does well with final probability .96
But CMsegment has final probability only .60 and an average amount of
practice less than 2
Suggestions for curriculum improvement: increase the amount of practice for
CMsegment
Intercept
slope
Avg
Practice
Opportunties
Initial
Probability
Avg
Probability
Final
Probability
CM

.15
.1
10.2
.65
.84
.92
CMarea

.009
.17
9
.64
.86
.96
CMsegment

1.42
.48
1.9
.32
.54
.60
Conclusions and Future Work
Learning Factors Analysis combines statistics,
human expertise, & combinatorial search to evaluate
& improve a cognitive model
System able to evaluate a model in seconds &
search 100s of models in 4

5 hours
Model statistics are meaningful
Improved models are interpretable & suggest tutor
improvement
Planning to use LFA for datasets from other tutors to
test potential for model & tutor improvement
END
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