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 1 1 1 1 1 1 2 1 1 1 2 1 1 2 1 2 2 1 1 2 1 2 2 1 1 4 1 2 2 2 1 1 4 1 5 2 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.