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Bayesian Modelling of Student Misconceptions in the one-digit Multiplication with Probabilistic Programming

Martin Ebner
Martin Ebner
Martin EbnerHead of Educational Technology at Graz University of Technology

Presentation at LAK conference 2016, Edingburgh April 2016

Bayesian Modelling of Student Misconceptions in the one-digit Multiplication with Probabilistic Programming

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W I S S E N T E C H N I K L E I D E N S C H A F T
http://elearning.tugraz.at
Bayesian Modelling of Student Misconceptions
in the one-digit Multiplication
with Probabilistic Programming
Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner,
Educational Technology & Institute of Theoretical Computer Science
Graz University of Technology
Edinburgh, 29.04.2016
http://elearning.tugraz.at
Presentation Outline
1. Main Idea
2. Previous Work
3. Model Structure
4. Learning the Model’s Parameters
5. Future Work
Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology
Edinburgh, 29.04.20162
http://elearning.tugraz.at
Main Idea
Model simple learning misconceptions
Probabilistic graphical models (bayesian networks)
Prediction of future student behaviour
How can this influence learning decision processes?
Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology
Edinburgh, 29.04.20163
http://elearning.tugraz.at
Previous Work (1/2)
Derive most prevalent error types in one-digit
multiplication problems
Explore the misconceptions and most difficult vs.
easy problems
Cluster one-digit multiplication problems according to
difficulty probabilities
Problems: No model, no adaptation, snapshot,
inflexible
Solution: Probabilistic Graphical Models
Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology
Edinburgh, 29.04.20164
http://elearning.tugraz.at
Previous Work (2/2)
1x1 Trainer: http://schule.learninglab.tugraz.at/einmaleins/ (last access 29 April 2016)
Developed by Graz University of Technology
Applied in different schools in Austria, Germany
and Switzerland
Limited information provided only by the answers,
no demographic values
Train the model (Model “learns” its parameters)
Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology
Edinburgh, 29.04.20165
http://elearning.tugraz.at
1x1 Trainer Application
Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology
Edinburgh, 29.04.20166

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Bayesian Modelling of Student Misconceptions in the one-digit Multiplication with Probabilistic Programming

  • 1. W I S S E N T E C H N I K L E I D E N S C H A F T http://elearning.tugraz.at Bayesian Modelling of Student Misconceptions in the one-digit Multiplication with Probabilistic Programming Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Educational Technology & Institute of Theoretical Computer Science Graz University of Technology Edinburgh, 29.04.2016
  • 2. http://elearning.tugraz.at Presentation Outline 1. Main Idea 2. Previous Work 3. Model Structure 4. Learning the Model’s Parameters 5. Future Work Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.20162
  • 3. http://elearning.tugraz.at Main Idea Model simple learning misconceptions Probabilistic graphical models (bayesian networks) Prediction of future student behaviour How can this influence learning decision processes? Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.20163
  • 4. http://elearning.tugraz.at Previous Work (1/2) Derive most prevalent error types in one-digit multiplication problems Explore the misconceptions and most difficult vs. easy problems Cluster one-digit multiplication problems according to difficulty probabilities Problems: No model, no adaptation, snapshot, inflexible Solution: Probabilistic Graphical Models Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.20164
  • 5. http://elearning.tugraz.at Previous Work (2/2) 1x1 Trainer: http://schule.learninglab.tugraz.at/einmaleins/ (last access 29 April 2016) Developed by Graz University of Technology Applied in different schools in Austria, Germany and Switzerland Limited information provided only by the answers, no demographic values Train the model (Model “learns” its parameters) Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.20165
  • 6. http://elearning.tugraz.at 1x1 Trainer Application Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.20166
  • 7. http://elearning.tugraz.at Difficult vs. Easy Questions Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.20167
  • 8. http://elearning.tugraz.at Average Time Consumption Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.20168
  • 9. http://elearning.tugraz.at Clustering One-Digit Multiplication Problems Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.20169
  • 10. http://elearning.tugraz.at Derived Error Types Error type Description e.g. 56 = 7 ∗ 8 operand errors a neighbouring operand is taken split 1 the neighbouring distance is 1 48 = 6 ∗ 8 split 2 the neighbouring distance is 2 for an operand or is 1 for both operands 40 = 5 ∗ 8 operand intrusions a digit of the result matches an operand first operand decade digit matches first operand 74 ← 7 ∗ 8 second operand unit digit matches second operand 68 ← 7 ∗ 8 consistency errors unit consistency only unit digit is correct 76 ← 56 decade consistency only decade digit is correct 51 ← 56 off-by- errors off-by-±1 the result differs by x = 1 55 or 57 off-by-±2 the result is off by −2 ≤ x ≤ 2 54, 55, 57, 58 pattern errors swapped digits in the result 65 confusion errors confusion with addition, subtraction and division operations 15 or 1 Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.201610
  • 11. http://elearning.tugraz.at Model Structure (1/4) Perturbation model Bug library with 6 error types : operand, intrusion, consistency, off-by-±1 and off-by-±2, pattern, confusion with addition, subtraction, and division operation Unclassified errors Correct (absence of misconception / error) Clean up the data samples Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.201611
  • 12. http://elearning.tugraz.at Model Structure (2/4) Minimal structural assumptions: Errors don’t depend on each other directly But there are answers to questions that could belong to many error types, f.e. a student may answer the question 8 × 5 with a 41, which could be both a consistency and off-by-+1 error. Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.201612
  • 13. http://elearning.tugraz.at Model Structure (3/4) Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.201613
  • 14. http://elearning.tugraz.at Model Structure (4/4) The learner’s state of knowledge changes over time even during application use The model represents our estimations about the student’s misconceptions The more data, the more certain we will be which error type is more dominant, how much, etc. “A particular person, when dealing with one-digit multiplication problems, makes 20% operand errors, 10% consistency errors” and so on Uncover the similarities and the differences of the learners Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.201614
  • 15. http://elearning.tugraz.at Learning the Model’s Parameters (1/3) Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.201615
  • 16. http://elearning.tugraz.at Learning the Model’s Parameters (2/3) The dependencies define the structure of the model The quantification of this relationships is expressed by the parameters, which will be learned by the data Initialisation assumption: non-informative uniform prior 1. Each error type and the unique correct option are equally likely 2. Given a specific error type, is there any preference or tendency towards a particular answer? Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.201616
  • 17. http://elearning.tugraz.at Learning the Model’s Parameters (3/3) Expectation-maximization (EM) - Algorithm, because the “LearningState” random variable is hidden Probabilistic programming library Figaro for Scala http://www.scala-lang.org https://www.cra.com/work/case-studies/figaro https://www.cra.com/sites/default/files/pdf/Figaro Tutorial.pdf https://www.manning.com/books/practical-probabilistic-programming Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.201617
  • 18. http://elearning.tugraz.at Future Work (1/2) Are there new types of errors (according to the model and data)? Identify groups of learners (stereotypes) with mixture of Dirichlet distributions (“Learning State” of 20% of our students has values near to a particular Dirichlet distribution, 30% equals a completely different Dirichlet distribution and so on) Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.201618
  • 19. http://elearning.tugraz.at Future Work (2/2) Predict the answer and use this to decide what the next learning issue will be Probabilistic and maximum a posteriori (MAP) queries can be used to influence all kinds of factors in a learning application such as hints, helping notes, rearrangement of questions’ sequence Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.201619
  • 20. http://elearning.tugraz.at Thanks for your attention! Comments? Behnam Taraghi, Anna Saranti, Robert Legenstein, Martin Ebner, Graz University of Technology Edinburgh, 29.04.201620