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

1

1
1

1
1 1

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1 1

1
2
1 1
2

1
2
1 1
2
2

1
2
1 1
2
4
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1 1
2
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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.
x X x

Geometry Problem: Diagram
One of the transversals bisects the segment of the other that is between the two parallel lines.
x x X x 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.
x x X x x

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.