Interactive Presentations of Mathematics

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Interactive Presentations of Mathematics

• Jeremy Gow Paul Cairns
• Computer Science
• Middlesex University, London

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Overview

• Why interactive proofs?
• Polya-Lamport framework
• The IMP system
• Interactive topology course
• Theory presentation
• Future plans

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Mathematical Proof

• Central to mathematics
• Mixture of prose and notation
• Deductive presentation
• Detail - incomplete, but convincing
• Context - extra-logical content

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Problems with Proof

• How much detail?
• Convincing
• Unintelligible and boring
• How much context?
• Helpful
• Justification
• Unnecessary and distracting
• You cant please everyone

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Using Hypermedia

• HTML, Java, XML, ...
• Reader controls proof presentation
• Aims
• Dynamic presentation of detail/context
• Easy to read/navigate
• Easy to write

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Polyas Four Stages

• Understand - terms, unknowns, diagrams
• Plan - abstract solution
• Execute - detailed solution
• Reflect - uses, alternatives, related theory
• Proof presentation format
• Argument vs. pre/post context

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Lamports Structured Proof

• How To Write A Proof
• Labeled hierarchy
• Explicit argument structure
• Reader chooses own depth
• Manages complex detail
• Exposes errors
• Ideal for hypertext

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Polya-Lamport Framework

• Interactive proof format (with hyperlinks)
• Statement of theorem
• Understand - unknowns, terms, diagrams
• Plan/Execute - structured proof
• Reflect - uses, alternatives, related theory
• Demo

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The IMP System

• Authoring toolkit
• XML proof grammar
• In Text proof file
• Explicit structure (XML)
• Maths notation (Latex)
• Out Expandable proof (Java applet)
• Need unified tool

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Interactive Topology Course

• Elements of Euclidean Metric Topology
  - Peter Collins
• Polya-Lamport framework
• Chapters, definitions, theorems
• Hand-coded HTML IMP applet
• Demo (www.cs.mdx.ac.uk/imp/topology)

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User Evaluation

• Seven topology students
• Tasks with course notes
• Questionnaires background, SUS
• Interview
• Results
• SUS average 78, range 63-88
• Design feedback

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Theory Presentation

• Chapters/definitions/theorems frames
• Shallow structure
• Is there a deep structure?
• Dialectic maths - Lakatos
• Hard to write/present
• Is it worth it?

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Future Plans

• Evaluation feedback
• GUI for XML proofs
• Summarise sub-proofs
• Develop theory presentation
• Large evaluation
• Automate sub-proof expansion?

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Summary

• Dynamic proof detail/context
• Polya-Lamport framework
• IMP XML to expandable proof
• Evaluated interactive course notes
• How to present theories?
• www.cs.mdx.ac.uk/imp