Analogical Solving
of Systems of Equations:
A Case Study
in Analogical Problem Solving
Svetlana Polushkina
October, 2006
Overview
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Goal
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Research questions
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Methods
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Experiment
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Model
Results and Discussion
Goal
Testing applicability
of Heuristic-Driven Theory Projection
to analogical problem solving
Figure 11
Figure
Research Questions
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Workings of an analogical problem solver
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Relation between analogy and structural
similarity
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Role of generalisations
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Useful background knowledge, heuristics and
similarity measure
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Appropriate representations
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Helpful hints
Methods
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Experiment with
human participants
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Manipulating provided
information
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Obtaining results
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Computational model
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Manipulating ...
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Obtaining ...
Comparing experimental and modelling results
Experiment Overview
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54 male and female pupils of the 8. grade of gymnasium
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Solving the target example problem
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Presented with the solution of the source example
problem
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Variants of the source problem presentation and
additional hints; 6 experimental conditions in total
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Data on time consumption and solution correctness
obtained
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Equal distribution of gender and math grades over
experimental groups
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Similarity assessments required immediately before and
after the problem solving
Similarity Assessment Questions
Experimental Manipulations
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Providing hints on mappings between
corresponding equations in the source and target
problems: Hint Factor
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Providing explanations on the derivations of the
source solution steps: Action Explanations Factor
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Equation 3 follows from equation 2 vs.
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Equation 3 follows from equation 2 by adding b
Providing explicit generalisations in the mapping
hints: Flat / Hierarchical Mapping Hint Factor
Experimental Results: Time Consumption
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Significant reduction in time consumption for the Hint Factor
(t(50) = 1.8183, p < .05)
No other effects on time consumption:
--> Automatic inferencing of solution steps producing actions
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Experimental Results
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Correct solutions produced by some participants in any
experimental group
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Significantly more structurally correct solutions
(77.78%), than just correct answers (48.15%),
(chi_square = 10.1647, p <.005)
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Interesting distribution of mistakes
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Confusing source and target values (all groups)
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Miscalculation (Hierarchical Mapping Hint groups)
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Misunderstanding of the 1. BF (Flat Mapping Hint groups, No
Mapping Hint groups)
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Confusing operators (No Mapping Hint groups)
Preference for structural similarity (87.96%) over
superficial similarity (12.04%)
–
Correlation between similarity assessment change and solution
correctness
Summary of the Experimental Results
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Give hints on mappings in educational contexts!
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Prefer structure-preserving mappings with
minimal information load
--> Such mappings can be produced
by Anti-Unification
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Use derivational example format to induce
analogical problem solving (and inferences on
step transitions) in educational contexts
Properties of the Model
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Employs higher-order anti-unification
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Produces structure-preserving least general
generalisations obeying to the systematicity principle
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Allows for flexible mapping
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Accounts for equivalence classes of algebraic
expressions (e.g. for problem re-representation)
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Keeps track of many aspects of the problem solving
process (step inputs and outcomes, step order, step
producing operators)
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Uses some heuristics and background knowledge
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Can solve relatively complex problems in a sub-domain
of mathematics
Similarity Measure
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Distance measure
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Costs for anti-unifying expressions
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1
3
7
Depth measure
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Rewards for structure preservation
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Same sorts / structures with different instantiations
Reversing arguments of a predicate
Applying a Structure Equalisation Rule
Anti-unifying arguments of a predicate
Applying a Structure Equalisation Rule
Cost (TOP) = 10000 / depth_value
5
1
Versions of the Model
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Providing some relevant anti-unification results to the
model: Hint Factor
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Mapping of the initial problems
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Mapping after re-representation during the application of the
1. BF
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Mapping after the application of the 1. BF
Excluding information on the actions undertaken to
produce the source solution steps from the step
descriptions: Actions Input Factor
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[Input, Outcome, Action] vs. [Input, Outcome]
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Versions inferring or just disregarding actions
Varying the type of processed generalisations: Flat /
Hierarchical Generalisation Outcome Factor
–
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Same processing, different format of stored results
6 versions in total, 5 fully functioning versions
Comparing Versions of the Model (I)
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Time consumption and number of necessary inferences for
different versions of the model were compared
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Significant reduction in time consumption for the Hint Factor
('masterWithProbe' query, t(18) = 176.79, p < .0005)
No other effects on time consumption of fully functioning model versions
Comparing Versions of the Model (II)
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Effects of the Hint Factor and the Flat / Hierarchical Generalisation
Outcome Factor on the number of necessary inferences
No effect of the Actions Input Factor on the number of necessary inferences.
Summary of the Modelling Results
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Give hints on mapping to facilitate problem
solving!
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Produce structure-preserving mappings and store
anti-unification results preferrably in a flat format
to reduce information load
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Keep track of the problem solving path (e.g.
already 'utilised' complex step producing
operators)
Experimental vs. Modelling Results
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Effect of the Hint Factor on
the time consumption
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No effect of the Action
Explanations Factor on the
time consumption
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No effect of the type of the
mapping hint on the time
consumption
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Miscalculation mistakes with
Hierarchical Mapping Hint
vs. understanding mistakes
with Flat Mapping Hint
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Effect of the Hint Factor on
the time consumption
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No effect of the Actions Input
Factor on the time
consumption of the successful
models
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No effect of the storage format
of anti-unification results on
the time consumption
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More inferences necessary
with Hierarchical
Generalisation Outcome
Format as with Flat GOF
Modelling results parallel experimental results
--> The model can be used to make conjectures about
analogical problem solving in humans
General Result of the Case Study
Heuristic-Driven Theory Projection
can be successfully used
to adequately model
analogical problem solving