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The Q-matrix method: A new artificial intelligence tool for data mining Dr. Tiffany Barnes Kennedy 213, tbarnes2@uncc.edu PhD - North Carolina State University

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Sep 10, 2004The Q-matrix method 2 Overview Introduction Adaptive Teaching and Data Mining Student Model Extraction Conclusions & Future Work

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Sep 10, 2004The Q-matrix method 3 Research challenge Turn the computer into a private tutor Diagnose and correct misconceptions Diagnosis tolerates careless errors & guesses Build a scientific approach to improving computer based education Build in fault tolerance, robustness Optimize for student performance Optimize teaching strategies for effectiveness

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Sep 10, 2004The Q-matrix method 4 The Problem Students take a tutorial and quiz online Determine what students know Redirect students to new/repeat material

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Adaptive Tutorial Flow Question Engine student Diagnostic Engine Concept Model Teaching Strategy Ask questions Student responds Determine learning path Determine concept state Select new material

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Sep 10, 2004The Q-matrix method 6 Assume contents affect behavior Data mining for knowledge student Contents Unknown Behavior Known

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Sep 10, 2004The Q-matrix method 7 Knowledge & student model Concepts Tutorial questions Student responses Student concepts Goal: Mine to extract student concepts

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Sep 10, 2004The Q-matrix method 8 Data mining & adaptive teaching Problem understanding Effective direction of student learning Data understanding Data from online tutorials Data preparation Select relevant variables Modeling: Q-matrix, cluster, factor Evaluation of results Misconceptions diagnosed? References: Data Mining Server @ http://dms.irb.hr/tutorial

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Sep 10, 2004The Q-matrix method 9 How the model works Student response 11100 Predicted responses: 01100 Err: 1 01101 Err: 2 11100 Err: 0 11111 Err: 2 Tutorial & Questions match 11100 Err: 0 Q-matrix 00011 10010 Student understands Concept 1 but not 2. Teaching Strategy

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Sep 10, 2004The Q-matrix method 10 How the model works-2 Concept state – a bit string that describes understanding Concept state 01: understands concept 2 but not concept 1 Q-matrix: concepts v. questions Each state has an “ideal response vector” computed from Q-matrix

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Sep 10, 2004The Q-matrix method 11 Binary Q-matrix example q1q2q3q4q5 Con1 0 0 0 1 1 Con2 1 0 0 1 0 Concept State IDR 00 01100 01 11100 10 01101 11 11111

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Sep 10, 2004The Q-matrix method 12 Research questions Are Q-matrix models interpretable? What factors affect Q-matrix extraction? How well does the Q-matrix method compare with other data mining methods?

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Sep 10, 2004The Q-matrix method 13 Results on simulated students Brewer tested 2 Q-matrix extraction methods based on ideal students + noise in ideal response vectors Q-matrix method needs few students for high noise tolerance, factor analysis needs many more References: Brewer 1996. NCSU Masters Thesis.

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Sep 10, 2004The Q-matrix method 14 Student model extraction Q-matrix, factor, and cluster models Compared for error on student data sets Q-matrix and cluster also compared by maps and by cluster convergence

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Sep 10, 2004The Q-matrix method 15 Q-matrix model Assumes concepts underlie questions Students are in “concept states” C: C1 = 1 understands concept 0 C2 = 0 doesn’t get concept 2 For each state, compute IDR Assign students to state with closest IDR

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Sep 10, 2004The Q-matrix method 16 Q-matrix creation Until convergence criterion met: 1. Increment number of concepts 2. Create random q-matrix 3. Fill concept states & compute error 4. Vary q-matrix 5. Fill concept states & compute error 6. Repeat steps 4-5 until error not improving 7. Repeat steps 2-6 to avoid local minima

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Sep 10, 2004The Q-matrix method 17 Factor analysis model Each tutorial question is a variable Create covariance matrix for vars Derive eigenvectors/values to explain most of the variance in the covar matrix Assumes that linear combinations of the variables will be able to explain the vars Eigenvectors ROTATED

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Sep 10, 2004The Q-matrix method 18 Cluster analysis model Answer vectors as points in plane Iterate until convergence: Choose random seed from data set Assign vectors to nearest seed Set new seeds to cluster medians Chooses random seeds, assigns vecs to closest seed, set new seed to cluster median Similar to q-matrix except seeds are Ideal Response Vectors

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Sep 10, 2004The Q-matrix method 19 Q-matrix vs. Factor Analysis CFA generated 4 factors/matrix Compared to q-matrix with 4 concepts Factor matrix converted to 0/1 Threshold of 0.3 -> 1, less -> 0 Factor matrix used as q-matrix Error computed for both Q-matrix performed significantly better (at least 19% less error/stud) on all 14 problems Smallest diff in performance when large amount of variance in student answers

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Q-matrix and factor errors per student

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Ratio of q-matrix to factor error and relative # of distinct observations

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Sep 10, 2004The Q-matrix method 22 Q-matrix vs. Cluster Analysis Cluster Analysis does not map to q- matrix as factor anal. does However, q-matrices do form clusters of students in the same concept state Ran Cluster Analysis with same number of clusters as q-matrix Similar clusters generated by both

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Sep 10, 2004The Q-matrix method 23 Clustering comparisons Determine equivalent concept state & cluster groupings (by largest overlap) These are in BOLD Count elements NOT in overlaps Overall diff = total NOT overlapping / total elements

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14,15 16 Con 0-4 Con 1-444 402,441,446,622 546,646,744 105,205,305 Con2-35 231 Con3-777 274 Proof 8 Q-matrix Cluster Comparison 6/15 clus different

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Differences in cluster overlap Ratio of different to total cluster assignments

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Sep 10, 2004The Q-matrix method 26 Q-matrix vs. Cluster Analysis 2 Each cluster has a “seed” Distances from seeds determine cluster membership For each cluster, summed differences between seeds & answer vectors Total error less than that of q-matrix clusters for all experiments

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Sep 10, 2004The Q-matrix method 27 Q-matrix vs. Cluster Analysis 3 Why is total error less for clusters? Because we force the IDRs in q-matrix method to be based on concepts This yields higher errors but more help in directing teaching strategies

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Sep 10, 2004The Q-matrix method 28 Q-matrix v. Clusters Summary If we used cluster results, how would we determine what to do for each student after the analysis? Cluster and q-matrix analyses could be used together for large data sets. Important: student outcomes

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Sep 10, 2004The Q-matrix method 29 Conclusions Full automation of economically expandable adaptive teaching system Method for diagnosis of misconceptions Q-matrix model interpretable by humans Q-matrix outperforms factor analysis in student modeling Q-matrix forms clusters similar to those in cluster analysis

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Sep 10, 2004The Q-matrix method 30 Future Work Any lesson can be augmented with diagnostic engine Different teaching strategies can be compared Apply Q-matrix method to benchmark data mining datasets Perform detailed time analysis and determine improvements Cross-validation tests to determine accuracy of model Missing data adaptations

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Sep 10, 2004The Q-matrix method 31 Thank you! Email: tbarnes2@nuncc.edutbarnes2@nuncc.edu This work was partially supported by NSF grants #9813902 and #0204222.

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Sep 10, 2004The Q-matrix method 32 How the model works-2 Student takes quiz Assigned to state with nearest IDR Error determined from difference between IDR & response, Q-matrix Q-matrices varied until error over all students is minimized

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Sep 10, 2004The Q-matrix method 33 Manual concept mapping Expert analysis of algebra tasks into rules Evolved into Q-matrix Relationship between questions & concepts Applications: Student assessment Group performance measure Finding new rules (student innovations) References: Birenbaum, et al. 1993, Tatsuoka 1983.

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Sep 10, 2004The Q-matrix method 34 Prediction of student data Hubal found that randomly generated rules were better predictors of student data than Tatsuoka’s Q-matrix This suggests that student data should be used to generate dynamic Q-matrices Mining for what the students know! References: Hubal 1992. NCSU Masters Thesis.

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Sep 10, 2004The Q-matrix method 35 Knowledge Assessment Comparison with expert models Remediation Tutorial effectiveness

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Sep 10, 2004The Q-matrix method 36 Remediation Analyze student states and apply a teaching strategy to direct next step Process: Find the least-understood concept, and have student retake the first lesson related to that concept

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Sep 10, 2004The Q-matrix method 37 Remediation results Self-guided choices compared with q- matrix choices Less than half of self-guided students chose differently Exam performance: q-predicted equal or worse than self-chosen Conclusion: remediation at least as good as student remediation

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