Download Why a Diagram is (Sometimes) Worth Ten Thousand Words

Why a Diagram is (Sometimes) Worth
Ten Thousand Words
Jill Larkin & Herbert Simon
Elodie Fourquet
CS888 Presentation
Outline
• Paper Overview
• Three Characteristics of Diagram:
– Localization
– Minimal Labeling
– Use of Perceptual Enhancement
• Extensions on Diagram
Problem Solving
• Reasoning
– Visual : Diagrams
Inherently spatial (indexed by plane location).
– Sentential : Aristotle philosophy
Inherently temporal (sequential/logical).
• In mind vs. on paper
Physics Problem
From a verbal physics problem, describing a pulleys & weights system:
• Using productions (sentential reasoning) is very complex.
• A better alternative exists for human: diagrams.
• Advantages of diagrams:
– localization &
– minimal labeling.
Pulley Problem: Diagram
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Diagram Advantages
• Indexing by location, minimal labeling.
• Adjacency of information.
∴ Shifts of attention are minimized.
Geometry Problem
Problem Statement:
“A pair of parallel lines is cut by a transversal...”
• Why is a diagram is most useful?
Geometry Problem: Diagram
“A pair of parallel lines is cut by a transversal...”
Geometry Problem: Diagram
“A pair of parallel lines is cut by a transversal...”
• Recognition ease: drawing reveals more. Angles appear.
• Visual hints on similar angles, recall Alternate Interior Angle axiom.
• Diagrams make use of Perceptual Enhancement.
Geometry Problem: Sentential
“A pair of parallel lines is cut by a transversal...”
• Recognition complexity:
no explicit mention of any angles relation in statement.
• Productions by direct translation do not contain angles.
• No Perceptual Enhancement.
Geometry Problem: Diagram
Two transversals intersect two parallel lines &
intersect with each other at a point X between the two parallel lines.
Geometry Problem: Diagram
Two transversals intersect two parallel lines &
intersect with each other at a point X between the two parallel lines.
X
Geometry Problem: Diagram
Two transversals intersect two parallel lines &
intersect with each other at a point X between the two parallel lines.
X
Geometry Problem: Diagram
One of the transversals bisects the segment of the other that is between
the two parallel lines.
Geometry Problem: Diagram
One of the transversals bisects the segment of the other that is between
the two parallel lines.
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Geometry Problem: Diagram
One of the transversals bisects the segment of the other that is between
the two parallel lines.
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X
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x
Geometry Problem: Diagram
Prove that the the two triangles formed by the transversals are
congruent.
Geometry Problem: Diagram
Prove that the the two triangles formed by the transversals are
congruent.
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Sentential
• Given a problem in English, we express it in a succinct form.
• Given a context, we know empirical rules, true relations.
Example: In Mechanics, Newton’s Laws.
• We develop a notation, a formal language that permits to use the
empirical rules in the specific problem.
• Productions : rules written using the established notation.
Sentential Example: Logic
• If Socrates is a human being, then Socrates is mortal.
Socrates is a human being.
Sentential Example: Logic
• If Socrates is a human being, then Socrates is mortal.
Socrates is a human being.
• Modus ponens
p→q
p
∴q
Sentential Example: Logic
• If Socrates is a human being, then Socrates is mortal.
Socrates is a human being.
• Modus ponens
p→q
p
∴q
∴ Socrates is mortal.
Production Example: User Interfaces
• Defining input problem: Propositional Production system.
• A production is: A set of conditions → a set of actions
%MouseDown, button==inactive → button=active,
!Repaint, !GrabMouseFocus
%MouseUp, button==active, ?In → button=inactive,
!Repaint, !ReleaseMouseFocus, ActionEvent>
%MouseUp, button==active, NOT ?In → button=inactive,
!Repaint, !ReleaseMouseFocus
Pulley Problem: Sentential
• Four productions from empirical rules.
Seven instances used to solve specific problem.
• Not logically complex, but almost impossible to solve.
• Lots in memory, constant search.
• Total elements searched: 138.
Geometry Problem: Sentential
• Problem statement has to be perceptually enhanced
Ex: segment, region, angles.
• In production rules, conditions have to be modified
Ex: alternate-interior-angle in terms of ‘parallel’,‘ between’, ‘region’,
‘side’...
Cost for recognition.
Diagrams Efficiency
• Localization.
• Minimal Labeling.
• Perceptual Enhancement.
Free-Body Diagram
William Playfair
Summary
• Two reasonings to solve a problem:
– sentences,
– diagrams.
• Diagram reduces search & augment recognition.
Diagrams contains explicit perceptual elements.
In Practice: Solving a Problem
• A figure to start, so to reason and intuitively solve the problems.
• Diagram: rough solution (often based on an instance).
• Proof by sequential worded arguments on the components using the
empirical laws.
• Words: valid solution (general deduction).
Beyond Diagram
• Diagram are static.
• Animation are dynamic diagram.
• ‘The Mechanical Universe’ and ‘The Mathematics Project’ of
Jim Blinn.
Beyond Diagram II
• Online environment for interaction.
• A system for creating and exploring mathematical sketches.
To Remember
• Diagram vs. Sentences.
• Cognitive Properties of diagrams:
– Localization
– Minimal Labeling
– Perceptual Enhancement.
• Animation: temporal diagram, dynamic visualization.
Final Remarks
• Words = laws vs. Diagrams = examples (special instances).
• What is easier for a computer = other problem (AI).
• First human reasoning for solving problem needs to be understood.
• Diagram/visualization can lower cognitive loads.
• Diagrams cannot solve everything.