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Dynamic Online Algebra Tutor: Enhancing Learning Through Cognitive Model Tracing

The MATHESIS project introduces an individualized web-based algebra tutor designed to enhance student learning through deep cognitive model tracing. By utilizing cognitive tutors inspired by the ACT-R theory, the system enables effective problem-solving and adaptive learning experiences. It provides step-by-step tutorial guidance, multiple solution paths, and individualized activity selection based on recorded cognitive skills. The software aims to improve efficiency and standardized test scores by offering advanced authoring tools to educators, thereby democratizing access to high-quality mathematics tutoring.

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Dynamic Online Algebra Tutor: Enhancing Learning Through Cognitive Model Tracing

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  1. An Individualized Web-Based Algebra Tutor Based on Dynamic Deep Model Tracing Dimitrios Sklavakis and Ioannis Refanidis dsklavakis@uom.gr, yrefranid@uom.gr Department of Applied Informatics Univercity of Macedonia Thessaloniki GREECE An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  2. Outline • The MATHESIS Project • Introduction: Cognitive Tutors • Motivation: Cognitive Tutors Successful Paradigm • Goals: Authoring Tools for Cognitive Tutors • Research approach: Bottom - Up • The MATHESIS Algebra Tutor • Web-based • Deep Cognitive Model Tracing • Broad Knowledge Monitoring • Related Work • Cognitive Tutor Authoring Tools (Carnegie Mellon) • Future Work • Ontology • Authoring Tools An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  3. The MATHESIS Project • Cognitive Tutors • Motivation • Goals • Research approach An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  4. The MATHESIS Project Cognitive Tutors • Model-tracing ITS build at Carnegie Mellon University • Learning by Doing: Problem-solving environment with interactive tools • Step by step tutorial guidance with feedback messages (correct, error, hints) • Can handle multiple solution paths • Adaptive problem selection and student pacing An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  5. The MATHESIS ProjectCognitive Tutors and the ACT-R theory • Adaptive Control of Thought-Rational: • Cognitive Theory of Learning and Performance • Learning by doing not by watching and listening • Cognitive Model Based on the ACT-R theory: • Problem solving knowledge is made of cognitive skills • A cognitive skill consists of: • Procedural knowledge: IF…THEN production rules • Declarative knowledge: Facts consisting of property-value pairs An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  6. Cognitive Tutor Technology:Use ACT-R theory to individualize instruction • Cognitive Model: A system that can solve problems in the various ways students can 3(2x - 5) = 9 If goal is solve a(bx+c) = d Then rewrite as abx + ac = d If goal is solve a(bx+c) = d Then rewrite as abx + c = d If goal is solve a(bx+c) = d Then rewrite as bx+c = d/a Hint: You must distribute a over bx and c Bug message: You must also multiply a by c Known = 85% Known =45% 6x - 15 = 9 2x - 5 = 3 6x - 5 = 9 • Model Tracing: The tutor matches the student’s steps against the solution produced by the cognitive model → context-sensitive instruction • Knowledge Tracing: The tutor records cognitive skill learning from problem to problem → individualized activity selection and pacing An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  7. The MATHESIS Project Motivation:Cognitive Tutors’ Real-world Success • Algebra Cognitive Tutor in over 2.000 schools in the USA, 300.000 students per year. • Geometry Cognitive Tutor in 350 schools • Approved by the U.S. Dept. of Education • Full year classroom experiments show significant efficiency gains: • 50-100% better on problem solving & representation use. • 15-25% better on standardized tests. An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  8. The MATHESIS Project Goal:Authoring Tools for Math Cognitive Tutors • Development costs of instructional technology are high • Approximately 300 development hours per hour of instruction for Computer Aided Instruction • Cognitive Tutors: • Approximately 200 development hours per hour of instruction • Requires PhD level cognitive scientists and AI programmers • Solution: Easy to use Cognitive Tutor Authoring Tools An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  9. The MATHESIS Project Approach:Bottom – Up Ontological Engineering The MATHESIS Authoring Tools: Guiding Tutor Authoring Through Searching in the Ontology The MATHESIS Ontology: Declarative description of the User Interface, Domain Model, Tutoring Model, Student Model and Authoring Model The MATHESIS Algebra/Math Tutor(s): Declarative and Procedural Knowledge hard-coded in a programming language Domain Experts’ Knowledge: Domain + Tutoring + Assessing + Programming An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  10. The MATHESIS Algebra Tutor • Web-based • User Interface: HTML + JavaScript • Specialized math editing applets: WebEq by Design Science • Declarative Knowledge: JavaScript variables and Objects • Procedural Knowledge: JavaScript functions • Domain cognitive model • Top-level skills (20) : algebraic operations (7), identities (5) , factoring (8) • Detailed cognitive task analysis gives a total of 104 cognitive (sub)skills • Detailed hint and error messages for all of the above An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  11. The MATHESIS Algebra Tutor • Tutoring model: deep cognitive model tracing through knowledge reuse When tutoring a cognitive skill, e.g. polynomial-multiplication the tutor traces the cognitive model for each one of the monomial-multiplications • Student model: broad knowledge monitoring • The tutor records and timestamps in a database the student’s performance for each skill that is tutored, giving a percentage assessment of cognitive skill learning over time • The tutor records in a database all the student’s interactions with the interface so that they can be re-traced at any time An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  12. MATHESIS Algebra Tutor Demo An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  13. Related WorkCMU Cognitive Tutor Authoring Tools • Example-tracing tutors: • Built through “programming by demonstration” • Authors create Examples of how the students should solve specific problems • For each solution step the author enters the answer • Cognitive Tutors • Built through Cognitive Task Analysis • Authors create Cognitive Models of how the students should solve a range of problems • For each solution step the author enters production rules • CTAT mainly supports Example-tracing Tutors An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

  14. Future Work • Ontological Engineering • Build a declarative description of the Algebra Tutor’s knowledge (Interface, Domain, Tutoring and Student models) • Build an Authoring Model through Cognitive Task Analysis of the Algebra Tutor creation • Authoring Tools • Search, Select, Modify the existing Ontology → Re-create (part of ) the existing Algera Tutor • Extend the Ontology → Create new Tutors! An Individualized Web-Based Algebra Tutor D.Sklavakis & I. Refanidis

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