Learning Factors Analysis – A General Method for Cognitive Model Evaluation and Improvement Hao Cen, Kenneth Koedinger, Brian Junker Human-Computer Interaction Institute Carnegie Mellon University

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 A Geometry Cognitive Model and Log Data Learning Factors Analysis algorithm 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 StudentStepSkillOpportunity Ap1s1Circle-area1 Ap2s1Circle-area2 Ap2s2Rectangle-area1 Ap2s3Compose-by-addition1 Ap3s1Circle-area3

Overview Cognitive Models & Cognitive Tutors Literature Reviews on Model Improvement A Geometry Cognitive Model and Log Data Learning Factors Analysis algorithm Experiments and Results

Learning Factors Analysis a set of factors that make a problem- solving step more difficult for a student Logistic regression, model scoring to fit statistical models to student log data A* search algorithm with “smart” operators for proposing new cognitive models based on the factors

The Statistical Model Probability of getting a step correct (p) is proportional to: - if student i performed this step = X i, add overall “smarts” of that student =  i - if skill j is needed for this step = Y j, add easiness of that skill =  j add product of number of opportunities to learn = T j & amount gained for each opportunity =  j p  Use logistic regression because response is discrete (correct or not) Probability (p) is transformed by “log odds” “stretched out” with “s curve” to not bump up against 0 or 1 (Related to “Item Response Theory”, behind standardized tests …)

Difficulty Factors Difficulty Factors -- a property of the problem that causes student difficulties  Like first vs. second parameter in LISP example above Four factors in this study  Embed: alone, embed  Backward: forward, backward  Repeat: initial, repeat  FigurePart: area, area-difference, area-combination, diameter, circumference, radius, side, segment, base, height, apothem Embed factor: Whether figure is embedded in another figure or by itself (alone) Example for skill Circle Area: Q: Given AB = 2, find circle area in the context of the problem goal to calculate the shaded area AB AB

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

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

The Split Binary Split -- splits a skill a skill with a factor value, & a skill without the factor value. StudentStepSkillOpportunity A p1s1 Circle-area-alone1 A p2s1 Circlearea-embed1 Ap2s2Rectangle-area1 Ap2s3 Compose-by- addition1 Ap3s1Circle-area-alone2 StudentStepSkillOpportunityFactor- Embed Ap1s1Circle-area1alone Ap2s1Circle-area2embed Ap2s2Rectangle-area1 Ap2s3 Compose-by- addition1 Ap3s1Circle-area3alone 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

Overview Cognitive Models & Cognitive Tutors Literature Reviews on Model Improvement A Geometry Cognitive Model and Log Data Learning Factors Analysis algorithm 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 tSlope Avg OpportuntiesInitial ProbabilityAvg ProbabilityFinal Probability Parallelogram- area Pentagon-area Student Intercep t student01.18 student10.82 student20.21 Model Statistics AIC3,950 BIC4,285 MAD0.083 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 1Model 2Model 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: 17 AIC: 3, BIC: 4, MAD: AIC: 3, BIC: 4, MAD: AIC: 3, BIC: 4, 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 1Model 2Model 3 Number of Splits: 4Number of Splits: 3Number of Splits: 4 Number of skills: 12Number of skills: 11Number 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: 3,884.95AIC: 3, AIC: 3, BIC: 4, BIC: 4, BIC: 4, MAD: 0.075MAD: 0.079MAD: 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 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 Results How about Compose-by-multiplication? InterceptslopeAvg Practice OpportuntiesInitial ProbabilityAvg ProbabilityFinal Probability CM With final probability.92 students seem to have mastered Compose-by-multiplication.

Beyond Experiments 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 InterceptslopeAvg Practice Opportunties Initial Probability Avg Probability Final Probability CM CMarea CMsegment

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

Acknowledgements This research is sponsored by a National Science Foundation grant to the Pittsburgh Science of Learning Center. We thank Joseph Beck, Albert Colbert, and Ruth Wylie for their comments.

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To do Reduce DFA-LFA.ppt, get from ERM lecture  Go over 2nd exercise on creating learning curves (from web site) in this talk & finish in 2nd session?  Print paper …. Other  Mail LOI feedback to Bett, add Kurt’s refs