Using Cognitive Modeling in Mathematics Instruction

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Steven Ritter, Founder and Chief Scientist at Carnegie Learning, presentation for the Cognitive Systems Institute Speaker Series on October 20, 2016.

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Using Cognitive Modeling in Mathematics Instruction

  1. 1. ©2014 Carnegie Learning, Inc. Using Cognitive Modeling in Mathematics Instruction Steve Ritter Carnegie Learning
  2. 2. ©2014 Carnegie Learning, Inc. Overview • Carnegie Learning background • Learning theory • Understanding student thinking • New directions
  3. 3. ©2014 Carnegie Learning, Inc. About Carnegie Learning • Started as a research project at CMU; company started in 1998 • Publisher of US K-16 Math Curricula – Primarily grades 6-12 – Blended implementation • Software, textbooks, Professional Development • Research-based, Proven effectiveness – US Dept of Education – WWC – 3rd party studies • Carnegie Mellon (and other university) research partnerships help drive product innovation • Approximately 600,000 students/year in 3000 schools (K12) – Nationwide
  4. 4. ©2014 Carnegie Learning, Inc. Cognitive Tutor principle • The more we understand about how students think and learn, the better we can help them think and learn
  5. 5. ©2014 Carnegie Learning, Inc. ACT-R • Complex knowledge is composed of simple knowledge components • Knowledge is strengthened through active use • Learning happens at knowledge component level John Anderson
  6. 6. ©2014 Carnegie Learning, Inc. MATHia Software Sequenced topics, unlocked upon completion Multi-step problem-solving Mastery via Bayesian Knowledge Tracing
  7. 7. ©2014 Carnegie Learning, Inc. § Model Tracing: § Tracks and provides feedback on individual student strategies § Provides Immediate feedback at each step through a solution § Diagnoses misconceptions leading to error and presents feedback to correct the misconception § Knowledge Tracing: § Tracks students growth in knowledge at a low level § Picks problems for each student, based on individual student needs § Zone of proximal development § Continuous formative assessment: – Each step is assessed and contributes to our knowledge about the student – Teacher reports emphasize areas that they can work on with students – Assessment is part of instruction Cognitive Model Cognitive Modeling
  8. 8. ©2014 Carnegie Learning, Inc. Cognitive Modeling Goals • Present complex problems – Knowing when to apply knowledge is just as important as having the knowledge • Design tasks to make thinking evident – Show your work • Allow multiple solution methods (when appropriate) • Associate steps with knowledge components • Trace learning on knowledge components
  9. 9. ©2014 Carnegie Learning, Inc. MODEL TRACING
  10. 10. ©2014 Carnegie Learning, Inc. What does this student understand about fractions? 5 1 x1 2
  11. 11. ©2014 Carnegie Learning, Inc. What does this student understand about fractions? Transcript: One half times one-fifth. Now, I have to find a multiple of 10. so half would go to five-tenths and one-fifth would go to two-tenths and multiply that and that would be one whole
  12. 12. ©2014 Carnegie Learning, Inc. Process • Multiply Fraction ( ) – Find common denominators • 1/2 = 5/10 • 1/5 = 2/10 – Apply operator to numerators • 5x2=10 – Keep common denominator • 10 – Simplify fraction • 10/10 = 1 € 1 2 × 1 5
  13. 13. ©2014 Carnegie Learning, Inc. Process • Multiply Fraction ( ) – Find common denominators • 1/2 = 5/10 • 1/5 = 2/10 – Apply operator to numerators • 5x2=10 – Keep common denominator • 10 – Simplify fraction • 10/10 = 1 • Add Fractions ( ) – Find common denominators • 1/2 = 5/10 • 1/5 = 2/10 – Apply operator to numerators • 5+2=7 – Keep common denominator • 10 – Simplify fraction • 7/10 € 1 2 × 1 5 1 2 + 1 5
  14. 14. ©2014 Carnegie Learning, Inc.
  15. 15. ©2014 Carnegie Learning, Inc. Conceptual Model
  16. 16. ©2014 Carnegie Learning, Inc.
  17. 17. ©2014 Carnegie Learning, Inc. KNOWLEDGE TRACING What is learned? Knowledge components!
  18. 18. ©2014 Carnegie Learning, Inc. Expression Writing
  19. 19. ©2014 Carnegie Learning, Inc. What gets learned?
  20. 20. ©2014 Carnegie Learning, Inc. Knowledge Tracing Students learn to solve positive slope problems at a different rate than negative slope problems Cognitive tutor traces these skills differently
  21. 21. ©2014 Carnegie Learning, Inc. RESULTS
  22. 22. ©2014 Carnegie Learning, Inc. Effectiveness at scale • RAND Corporation, with funding from US Dept. of Education (IES) • Algebra 1, blended implementation • Random assignment of 147 schools, 19,000 students • 7 regions across US • 2 cohorts • Intent-to-treat analysis • Results from year 2 HS • No diff in year 1
  23. 23. ©2014 Carnegie Learning, Inc. FUTURE DIRECTIONS
  24. 24. ©2014 Carnegie Learning, Inc. Cognitive Model Improvement • Use student data – Refine skill model (Learning Factors Analysis) – Split and merge skills • Fit Bayesian Knowledge Tracing Parameters • Improve task design
  25. 25. ©2014 Carnegie Learning, Inc. Understanding implementation Classes that violate mastery learning do worse over time Implementations improve over time.
  26. 26. ©2014 Carnegie Learning, Inc. Assessment • Adaptive Personalized Learning Score (APLSE) • Formative assessment used for summative purposes • Replace high-stakes tests • Enables • Assessment of problem solving, complex tasks • More instructional time • More educationally effective practices • Less test anxiety, test-taking skills • No test surprises
  27. 27. ©2014 Carnegie Learning, Inc. Questions

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