1 Domain Knowledge, Structural Learning Theory & Role in Building Teaching and Learning Systems April 10, 2006 Symposium on Knowledge Representation TICL SIG Joseph M. Scandura, Ph.D. Chairman, Board Scientific Advisors, MERGE Research Institute Emeritus and Adjunct Professor, University of Pennsylvania Visiting Research Professor, College of Information Science and Technology, Drexel University www. scandura.com It’s hard to believe 36 years have gone by since I first introduced the SLT at the 1970 SL Conference in Philadelphia, repeated a couple days later here. A lot has gone on since then, but the focus has always been on understanding fundamentals – on four basic questions.
Transcript
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Domain Knowledge, Structural Learning Theory & Role in Building Teaching and Learning Systems
April 10, 2006
Symposium on Knowledge RepresentationTICL SIG
Joseph M. Scandura, Ph.D.Chairman, Board Scientific Advisors, MERGE Research Institute
Emeritus and Adjunct Professor, University of Pennsylvania
Visiting Research Professor, College of Information Science and Technology, Drexel University
www. scandura.com
It’s hard to believe 36 years have gone by since I first introduced the SLT at the 1970 SL Conference in Philadelphia, repeated a couple days later here. A lot has gone on since then, but the focus has always been on understanding fundamentals – on four basic questions.
It’s hard to believe 36 years have gone by since I first introduced the SLT at the 1970 SL Conference in Philadelphia, repeated a couple days later here. A lot has gone on since then, but the focus has always been on understanding fundamentals – on four basic questions.
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Research Motivated by Four Basic Questions
Content: What does it mean to know something? Specifically, how can competence (content knowledge) be represented so it is executable & has direct behavioral relevance?
Assessing Behavior: How can one determine individual knowledge? What does an individual know and not know about any given content?
Cognition: Why can some people solve problems whereas others cannot? What are the basic mechanisms & constraints governing how learners use and acquire knowledge?
Instruction: How does knowledge change over time as a result of interacting with an external environment?
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Cognitive Models in Teaching & Learning (TICL)
Top-down: Cognitive Models in TICL provide motivation & guidelines for TICL systems
Bottom-up: Extend AI &/or Learning Theories to support TICL
Goal of Structural Learning Theory (SLT): Fill Gap between high level conceptualization & executable systems
Like most cognitive models, SLT started at the top (like cognitive models but with deterministic assumptions)
Continuing refinement & extension has made SLT fully executable for the first time (like AI & biologically inspired models & theories but with behavior/observable emphasis)
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Overview of Structural Learning Theory w/ Authoring & Delivery Systems Needed for Automation
I-A. Content Knowledge Representation
tasks/problems lower & higher order SLT rules
TutorITI-A. Content knowledge w/
III. UCM, capacity/speedIV. Full diagnostic & tutorial
I. Representing Observables & Knowledge as SLT Rules
• I-A. Content Knowledge (Competence) Represented at Multiple Levels of Abstraction as SLT Content Rules
• SLT Rules include both Structural & Procedural Abstract Syntax Trees (ASTs)
• Structural/Declarative ASTs of SLT Rules• Represent Domain & Range Data Structures• Correspond to Perceptual/Automated Knowledge
• Procedural ASTs of SLT Rules• Represent Hierarchies of Behaviorally Equivalent Processes• Correspond to Procedural Knowledge
• I-B. Observable Behavior Represented as Problem ASTs
• Represent Observables (e.g., problems)• Via which Learners & Tutors Interact
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Sample Problem AST for
hundreds
Top= 7
Given AST (Initialized Nodes Domain AST)
Goal AST
tens ones
differenceBottom= 5
borrow digit
Bottom= 2
differenceborrow digit
borrow digit differenceTop= 5
Top= 0
Bottom= 9
705-529
problem
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Abstract Syntax Tree (AST) Definition of SLT Rules
category
Domain-Range AST Procedure AST
SLT Rule
component
dynamic
loop
condition sequence
SLT Rule
Domain
prototype
operationIF..THEN
Range
Procedure
refinement types
…
component
SL
T R
ule
SKIPSKIP
Dynamic & Interaction
refinements
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component
Abstract Syntax Tree (AST) SLT Higher Order Rule for
Column Subtraction
Domain-Range AST Procedure AST
prototype
category dynamic
SLT Rule
loop
condition sequence
SLT Rule
Domain
subtract (top, bottom: ;
difference)
IF..THEN
Range
Procedure
refinement types
…
component
…
SL
T R
uledraw difference
digit –e.g., 5
10
component
Abstract Syntax Tree (AST) SLT Higher Order Rule for
Docking Space Station
Domain-Range AST Procedure AST
prototype
category dynamic
SLT Rule
loop
condition sequence
SLT Rule
Domain
fire_rocket (start, end, time: ;
movement)
IF..THEN
Range
Procedure
refinement types
…
component
…
SL
T R
ulefire_rocket
(start, end, time:..;distance)
BRIEFBRIEF
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Structural Knowledge Input-Output Data Structure AST defining Column Subtraction
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Procedural Knowledge Procedure AST Generating Specified Input and Output Behavior
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ALL Content Knowledge represented by Sets of SLT Content Rules
• Behavior is represented as Problems; Knowledge as SLT Content Rules(domain dependent & independent; declarative & procedural; h.o. &l.o.)
SLT Content Rule = AST structure & procedure, representing multiple levels of equivalent knowledge; Behavior associated with various levels is equivalent but not identical
Individual SLT Rule = slice (single level) in an SLT Content Rule* individual differences in mastery level: represented by specific levels of abstraction in ASTs
defining Individual SLT rules declarative knowledge: Procedure is simple (e.g., top-level); Structure is correspondingly
complex.* procedural knowledge: Structure is simple (e.g., top-level); Procedure is correspondingly complex.
*Note: Multiple gradations between declarative & procedural knowledge
higher order knowledge/meta-knowledge/heuristics/deduction: Structure of SLT (h.o.) Rule includes other SLT Rules. H.O. rules generate new SLT rules
conflict resolution/rule selection/design alternatives: H.O. rules select from alternative rules (e.g., design)
Structural (Content) Analysis (SA):Summary & Benefits I
• Early Research* Showed that Identifying Expected Behavior & What Must be Learned made Empirical Research Largely Redundant
• Result Motivated Development of a Systematic (now Patented) Process for Knowledge Representation associated with any Given Domain
Roughead, W.G. & Scandura, J.M. “What is learned” in mathematical discovery. Jr. Educational Psychology, 1968, 59, 283-298.
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II. Structural Analysis (SA):A Cognitive Meta-Theory
A Systematic, Extensible & Patented Method for Subject Matter Experts (SME) to Represent Observable Behavior & Knowledge as AST-based Problems & SLT Content Rules
1. Start with Informally Defined Problem Domain: Select & Systematically Define Representative Sample of Prototypic Problems in Domain & Represent in Terms of ASTs
2. Systematically Construct SLT Rules for Solving Prototypic Problems
3. Convert SLT Rules into Higher Order Problems
4. Construct Higher Order SLT Rules for Solving H.O. Problems
5. Optionally Eliminate Redundant SLT Rules
6. Repeat Process Until Desired Level of Domain Coverage Is Attained
1. SME Selects & Represents Well-Defined Problems as Hierarchical ASTs
Whole Number Arithmetic ___ 4027 324 324 37 | 285
- 2535 256 x 37+ 37
Domain of Bedrooms to be CleanedBedroom <presentable, unpresentable>Bed <made, unmade>Carpeting <clean, dirty>Rug1 <clean, messy, messy-dirty>Rug2 <clean, messy, messy-dirty>Rug3 <clean, messy, messy-dirty>
One SLT Solution Rule Sufficient to Solve each Problem Type SLT solution rules also can be represented with any desired degree of precision (because ASTs may be refined arbitrarily)
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Problem Structure (AST) Problem Layout Node Attributes
1. Sample Problem in AuthorIT Input-Output ASTs for Mixed Fractions
Proofs in High School Trigonometry Examples: sin2 A + cos2 A = 1 -- ?proof
a2 + b2 = c2 -- ? prooftan2 A + 1 = sec2 A -- ? proof
Key is for SME to select only representative problems i.e., intuitively different problems – problems requiring different kinds of representations &/or solution methods SME can represent problems with any desired degree of precision
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Simple Ill-Defined Domains(emphasis on identifying SLT rules & h.o. rules)
2. Construct Solution Rules for Prototypic ProblemsDomain of measure conversion problems Example 1A: yd 36_times in Example 1B: gallons 8_times pints
Domain of number series problems* Example 2A: 1 + 3 + 5 + … + 99 50x50 2500 Example 2B: 1 + 3 + 5 + … + 99 50x(1+99)/2 2500 Example 2C: 1 + 3 + 5 + … + 99 successive addition 2500
Proofs in High School Trigonometry Example 3:
sin2 A + cos2 A = 1 start with a2 + b2 = c2, divide by c, substitute sin, cos definitions
Proof is resulting steps_____* For early research on this subject see:Scandura, J.M., Woodward, E., & Lee, F. Rule generality and consistency in mathematics learning. American Educational Research Journal, 1967, 4, 303-319.Scandura, J.M. Learning verbal and symbolic statements of mathematical rules. Journal of Educational Psychology, 1967, 58, 356-364.
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Simple Ill-Defined Domains 3. Convert SLT Rule to Higher Order Problem
(Construct Goal & Given of Higher Order Problem)A. Replace semantic-specific nodes in Solution Rule with abstractions. B. Select given rules from which Solution Rule can be constructed.
Example 1Problem: 3 yd -- ?in
Solution Rule: yd 36_times in
Higher Order Problem:
Givens: yd n1_times xxxxxx n2_times in
Goal: blug n_times clug
i) blug & clug = units of measurement ii) n is a specific number
iii) variations include substituting “op” for “times”
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Simple Ill-Defined Domains 3. Convert SLT Rule to Higher Order Problem
(Construct Goal & Given of Higher Order Problem)A. Replace semantic-specific nodes in Solution Rule with abstractions. B. Select given rules from which Solution Rule can be constructed.
Example 2 Example 2A: 1 + 3 + 5 3x3 9
1 + 3 + 5 + … + 2n-1 nxn Sum
Example 2B: 1 + 3 + 5 3x(1+5)/2 2500 a + a+d + a+2d + … + 1 n x (a + l)/2 Sum
Example 2C: 1 + 3 + 5 successive addition 2500
a1 + a2 + a3 + … + an-1 successive addition Sum
n = no. termsa/l/d = first/last term/common differenceai = arbitrary term in arithmetic series
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Simple Ill-Defined Domains 3. Convert SLT Rule to Higher Order Problem
(Construct Goal & Given of Higher Order Problem)A. Replace semantic-specific nodes in Solution Rule with abstractions. B. Select given rules from which Solution Rule can be constructed.
Example 3sin2 A + cos2 A = 1
start with a2 + b2 = c2, divide by c, substitute sin, cos definitions Proof is resulting steps
trig identity start with a2 + b2 = c2, divide by side, substitute trig definitions Proof is resulting steps
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4. Construct SLT Higher Order Rule to Solve Higher Order Composition Problems
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4. Alternative SLT Higher Order Rules to Solve Higher Order Generalization Problem
• Higher Order SLT Rules* Example 2A:
1 + 3 + 5 3x3 9
a1 + a2 + a3 + … + an-1 nxn Sum replace three terms by n
Example 2B:
1 + 3 + 5 3x(1+5)/2 9 1 + 3 + 5 + … + 2n-1 n x (a + l)/2 Sum
replace 1 by a, 5 by l &/or three terms by n Example 2C:
1 + 3 + 5 1+3+5 9 a + a+d + a+2d + … + 1 successive addition Sum
replace each term by a variable, three terms by n________* In these examples, “1 + 3 + 5” may be ANY specific arithmetic series
GivenGiven
GoalGoal
ProcedureProcedure
GivenGiven
GoalGoal
ProcedureProcedure
GivenGiven
GoalGoal
ProcedureProcedure
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4. Different examples result in Different generalizations with
Different domains of applicability*replace number of terms by n & multiple n x n [very efficient but works only with arithmetic series beginning with 1 with a common difference of 2]
replace number of terms by n, first by a, last by l and compute n (a+l)/2[efficient; works with ALL arithmetic series]
replace each term by a variable & add successively[very inefficient but works with ALL series, arithmetic or otherwise]
___________*Scandura, J.M., Woodward, E., & Lee, F. Rule generality and consistency in mathematics learning. American Educational Research Journal, 1967,
4, 303-319.Scandura, J.M. Learning verbal and symbolic statements of mathematical rules. Journal of Educational Psychology, 1967, 58, 356-364.
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5-6. Eliminate Redundant Solution Rules
5. Higher order rule may generate solutions for any number of problems of similar type • Kernel of truth truth behind typologies (cf. Polya, 1962; Scandura, M:CBF,
1971; Jonassen, Spector & others in Y2K)• New conversion rules generated as needed from basic rules; Basic rules
can be added at will• e.g., 12 ft. = 12 in., 4 qt. = 1 gal., etc.
• Hence, Original solution rules become redundant• i.e.,derived as needed via higher & lower order rules
6. Process can be continued indefinitely• Convert new rules to still higher order problems, etc.• Procedures in enhanced rule set become simpler but generating power
goes up dramatically expanding coverage in original domain
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4-5-6. Higher Order Selection Rules (a/k/a Conflict Resolution)*
1 + 3 + 5 3x3 9 1 + 3 + 5 + … + 2n-1 nxn Sumreplace three terms by n
1 + 3 + 5 3x(1+5)/2 2500 a + a+d + a+2d + … + 1 n x (a + l)/2 Sum replace 1 by a, 5 by l &/or three terms by n
1 + 3 + 5 successive addition 2500
a1 + a2 + a3 + … + an-1 successive addition Sum replace each term by a variable, three terms by n
One Higher Order Selection Rule:
Case Type-of-Series: a) starts with 1 with a common difference of 2, select rule N x Nb) common difference, select rule N x (A+D)/2 c) else select successive addition
A more General but Error-prone Selection Rule:
Choose the simplest rule_____*Importance of selection rules becomes clear In discussion of associated SLT theory.
Con
flic
tin
g R
ule
s
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Kinds of Higher Order SLT rules:Schematics
Composition: A --> B, B --> C ==> A --> B --> C
Analogy: A1 --> B ==> A2 --> B
Generalization: A0 --> B0 ==> A --> BSelection: A --> B, B --> C ==> A --> B or
B --> C Automation:A1, A2 --> B1, B2 ==> A --> B
where A = parent of A1, A2B = parent of B1, B2
Retrieval OthersCombinations
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Structural Analysis (SA):Summary & Benefits
• SA Systematic: Method of SA is highly systematic• SA partially automated with much of remainder automatable
• SA Indefinitely Precise: Advance in AST (hierarchical) representation makes level of detail arbitrary
• High level conceptualization thru atomic representation possible• SA can be continued indefinitely as desired• Domain of applicability is automatically specified by AST structures
(in SLT content rules)
• SA Universally Applicable: Applicable to arbitrarily complex domains
• Domain coverage indefinitely extendable• New higher (& lower) order rules automatically introduced as needed • SA cumulative – builds on prior SA
• Generating Power increases monotonically• SLT rules tend to become simpler as SA continues• (breadth of) coverage & collective generating power goes up qualitatively
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Structural (Content) Analysis (SA)
What is Learned?
Bad NewsSA of Content requires work
Good News Experience shows SA adds precision & minimizes need for empirical research Preliminary SA is helpful Further SA is better Atomic SA is best
SA is cumulative one can build on preliminary SA without loss
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Structural AnalysisFoundation for Structural Learning Theory (SLT)
Structure of a KR alone is sufficient to guide T &L
SLT builds on structural (content analysis) to: assess lower & higher order knowledge predict behavior specify needed instruction
with arbitrary degrees of precision
SLT – a general & precise infrastructure for automated learning & tutoring systems
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Cognitive Theory: Transitions from
Naive to Neophyte to Master
Why is it that some people can solve problems that others cannot? And, how is it that initially naïve learners acquire new competence? And, gradually come to acquire mastery associated with experts?
(Quote from Scandura, 1981, Educational Psychologist, p. 139)
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III. Structural Learning Theory SLT - Cognitive Theory: Universal Control Mechanism,
Processing Capacity, Processing Speed
I-A. Content Knowledge Representation
tasks/problems lower & higher order SLT rules
TutorITI-A. Content knowledge w/
III. UCM, capacity/speedIV. Full diagnostic & tutorial
Universal Control Mechanism (UCM) How Rules are Used & New Ones Generated (A Least Common Denominator with Minimal Assumptions)
Overview of a Patented Method*
• Check available rules to see which AST structures match the given problem
• Unless exactly one SLT Rule matches, control goes to a deeper level looking for rules whose ranges contain structures that match the given problem (a recursive process)
• Once exactly one rule is found, that rule is applied & new rule generated
• Control reverts to previous level & process continues with checking at previous level of embedding
• Eventually, process halts because problem is solved or processing capacity is exceeded (alternatively a predetermined recursion limit may be set in automated systems)
* See Figs. 27-27A in U.S. Patent 6,275,976
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bedroom {presentable} bed {made} carpet {clean}
Example of UCM in Action:Initial Problem and Partial SLT Rule Set
--?
Initial Problem
bedroom {not-presentable} Component bed {unmade} carpet {dirty}
Original SLT Rule setmake bed (DOMAIN) make (bed) (PROCEDURE) bed (RANGE)vacuum carpet (DOMAIN) vacuum (rug) (PROCEDURE) carpet (RANGE)
_____No Lower Order SLT rule in Rule Set matches problem.Hence, control seeks rules whose range includes rules that do match
Lower Order SLT Rules in (Partial) Rule Set
?
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Example of UCM in Action:Higher Order SLT Rule
Apply SLT-rule1 and SLT-rule2 in parallel {parallel refinement}
_____1. Range Structure of Higher Order Rule matches Problem Structure2. Control seeks to match H.O. Rule Domain against set of available SLT rules 3. Domain of higher order rule satisfied by lower order SLT rules in rule set
Range of Higher Order Conjunction Rule
Domain of Higher Order Conjunction Rule
Procedure of Higher Order Conjunction Rule
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Example of UCM in Action:Higher Order SLT Rule Generates New Solution Rule
Result:
1. Higher Order SLT (Conjunction) Rule is applied to make & vacuum SLT rules in Rule Set.
2. Newly generated solution rule clean is added to set of available rules
3. Control checks original problem against rule set enhanced w/ clean
4. Control reverts to previous level where newly generated rule, clean, matches, is applied & solves original problem
clean (bedroom) make (bed) vacuum (carpet)
Newly Generated Solution Rule
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Importance of Universal Control Mechanism (UCM)?
• Empirical Research Supports UCM & Processing Constraints
• UCM Available from Earliest Ages (e.g., JEP, Sam)• Fixed Processing capacity (Voorhies)• Processing Speed (observation) • Emphasizes Observable Behavior Not Brain Physiology
• Applicability to both Human Behavior & Automated Intelligence
• Supports Incremental Development of Knowledge Base
• Continuing SA introduces new SLT rules as needed
Ability to Add Learning, Conflict Resolution & Chunking SLT rules without change to UCM
Supports ill-defined problem solving, design (selection) & automatization without change
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IV. Structural Learning Theory Diagnostic and Instructional Logic
I-A. Content Knowledge Representation
tasks/problems lower & higher order SLT rules
TutorITI-A. Content knowledge w/
III. UCM, capacity/speedIV. Full diagnostic & tutorial
expertise;fully configurable
LearnerIII. U Control Mechanism,
capacity/speedIV. Individual knowledge
II. Structural Analysis viaAuthorIT AutoBuilder Blackboard Editor TutorIT Options
Diagnosis = Assessing Behavior PotentialDetermining Known & Unknown Parts of SLT Rules
Examples: subtract with borrowing (but not with zeros in top); adding ‘ing’ to verbs with silent ‘e’ (but not when verb ends in consonant)
Given a problem, patented processes show how an SLT solution rule implicitly & automatically defines a set of diagnostic sub-problems
These sub-problems correspond to nodes (at various levels) in the defining procedural AST
Assuming Sufficient Precision (i.e., atomic refinement) Research shows that a Single Test Item under Atomicity conditions is Sufficient to Determine Whether the Learner Knows the corresponding Node
Learner’s Current State of Knowledge wrt SLT rule is Represented by Assigning +, -, ? to Nodes
Probabilities or multiple test items may be used when analyses are incomplete
Distinguishing Atomicity Level in SLT Rule Hierarchies
higher levels in hierarchy have less detailed processes & more complex structures: top level corresponds to atomic rules equivalent to declarative knowledge (faster execution)
Procedural Steps at a Lower Level in AST Hierarchy
4 3 4 3 4 3 4 3 4 3-2 7 -2 7 -2 7 -2 7 -2 7
6 1 6
Procedural Steps at the Top Level in AST Hierarchy
4 3 4 3-2 7 -2 7 (e.g., working problem in head)
1 6
/13
1
/13
1
/3
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Assessing Behavior PotentialHigher Order Knowledge*
Assessing Higher Order SLT rules Requires Problems in which Givens and/or Goals include Processes (other SLT rules)
A B, B C ==> A B C
In Complex Domains: It is Sufficient to Assess Behavior on Rules and Higher Order Rule Individually
Universal Control Mechanism makes it Possible to Predict Behavior on Complex Problems whose Solution Requires both Higher and Lower Order SLT rules
* Scandura, J.M. Role of higher order rules in problem solving. Journal of Experimental Psychology, 1974, 120, 984-991.
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TutoringInfluencing What a Learner Knows
Deciding What to Teach and When to Teach: Based Entirely on the Structure of SLT Rules to be Learned
Learner’s Current State of Knowledge wrt SLT rule is Represented by Assigning +, -, ? to Nodes
Standard Pedagogy: If Learner’s Status on Node is Undetermined (?) Test Unknown (-) Teach if Prerequisite Nodes is Mastered Known (+) Select Next Node in Execution or Mastery (+ w/ latency) add time constraints
Other Pedagogies Range from making Larger (or smaller) Leaps (e.g., teaching when when prerequisites
undetermined and/or selecting nodes from top-down) to fully Learner Controlled
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Quick Summary of SLTStructural (Content) Analysis: systematically identify desired behavior & what must be learned:
prototypic problems represented hierarchically as AST-structures with Givens & Goals knowledge represented hierarchically via AST-based SLT content rules higher order & selection rules systematically identified;
play a key role in ill-defined & design problem solving
Cognition: SLT rules & higher order rules plus control & processing universals
Diagnosis & Instruction: diagnostic sub-problems & instruction associated with AST nodes of SLT rules individual knowledge & needed instruction based on performance on sub-problems defined
by AST nodes current state of individual’s SLT rule knowledge & pedagogical logic determine instruction at
each point in time h.o. rules used to assess extra-domain problem solving, rule selection/motivation & mastery transition from naïve to neophyte to master, with mastery opening possibilities for new levels
of learning
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SLT Problem(s) & Set(s) of Higher & Lower Order Rules
Extension to Multiple Learners AST Knowledge Representation, Human Interface &
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