using cognitive modeling in mathematics instruction
TRANSCRIPT
©2014 Carnegie Learning, Inc.
Using Cognitive Modeling in Mathematics Instruction
Steve RitterCarnegie Learning
©2014 Carnegie Learning, Inc.
Overview
• Carnegie Learning background• Learning theory• Understanding student thinking• New directions
©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
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Cognitive Tutor principle
• The more we understand about how students think and learn, the better we can help them think and learn
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ACT-R
• Complex knowledge is composed of simple knowledge components• Knowledge is strengthened through active use• Learning happens at knowledge component level
John Anderson
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MATHia Software
Sequenced topics, unlocked upon completion
Multi-step problem-solving
Mastery via Bayesian Knowledge Tracing
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§ 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
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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
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MODEL TRACING
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What does this student understand about fractions?
51x12
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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-tenthsand one-fifth would go to two-tenthsand multiply that and that would beone whole
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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
€
12×15
©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
€
12×15
12+15
©2014 Carnegie Learning, Inc.
©2014 Carnegie Learning, Inc.
Conceptual Model
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©2014 Carnegie Learning, Inc.
KNOWLEDGE TRACINGWhat is learned? Knowledge components!
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Expression Writing
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What gets learned?
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Knowledge TracingStudents learn to solve positive slope problems at a different rate than negative slope problems
Cognitive tutor traces these skills differently
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RESULTS
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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
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FUTURE DIRECTIONS
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Cognitive Model Improvement
• Use student data– Refine skill model (Learning Factors Analysis)– Split and merge skills
• Fit Bayesian Knowledge Tracing Parameters
• Improve task design
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Understanding implementation
Classes that violate mastery learning do worse over time
Implementations improve over time.
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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
©2014 Carnegie Learning, Inc.
Questions