Designing a Proof GUI for Non-Experts
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Designing a Proof GUI for Non-Experts

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Results fron an experiment and brainstorm on how to design a GUI for teaching mathematical proofs.

Results fron an experiment and brainstorm on how to design a GUI for teaching mathematical proofs.

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  • 1. Designing a Proof GUI for Non-Experts Evaluation of an Experiment Martin Homik, Andreas Meier Presentation by Christoph Benzmüller UITP 2005, Edinburgh ActiveMath Group German Research Center for Artificial Intelligence DFKI GmbH, Saarbrücken
  • 2. Motivation
    • Typical proof GUI design:
    • Proof system centered
    • Too specific; For experts only
    • Non-Expert proof GUI design:
    • User centered
    • Deliver what the user needs!
  • 3. Motivation (2)
    • MIPPA Project goals :
    • Interactive learning tool for math. proof
    • Underlying proof engine:
    • Proof planner MULTI
    • Target group:
    • Undergraduate students
    • A-level pupils
  • 4. Expert GUI: Loui
  • 5. Towards a User Centered GUI
    • First step:
    • Paper&Pencil student experiment
    • Primary task:
      • Observe basic user wants and needs
  • 6. Experiment Setting
    • 4 Groups
    • 2 students in each
    • Background :
    • Computer Science, Math, Logic
    • No design restrictions:
    • creativity/underlying system
    • use/invent functionalities freely
    • Example Theorem:
    • Irrationality of √2
    • Use :
    • Definitions
    • Term rewriting
    • Island introduction
    • Contradiction
    Design (120 min) Presentation (15 + 10 min) Discussion
  • 7. Experiment Remarks
    • This is no HCI experiment:
    • We let users design.
    • Users were already familiar with PP/Loui.
    • Users were restricted to certain tasks.
    • Why?
    • First attempt: obtaining inspiration
  • 8. Textbook Example: √2 is irrational „ Assume that √2 is rational. Then, there are integers n,m that satisfy √2= n / m and that have no common divisors. From √2= n / m follows that 2* m 2 = n 2 (1), which results in the fact that n 2 is even. Then, n is even as well and there is an integer k such that n =2* k . The substitution of n in (1) by 2* k results in 2* m 2 =4* k 2 which can be simplified to m 2 =2* k 2 . Hence, m 2 and m are even as well. This is a contradiction to the fact that n,m are supposed to have no common divisor.“
  • 9. Group A: Text-based Textual presentation of a proof. The same way as taught at school.
    • There exist no two integers m and n:
      • m and n being coprime
      • √ 2 =m/n
    √ 2 is irrational check proof complete proof automatically feedback no logical notation √ 2 is irrational Statement access
  • 10. Group A: Operator Application
    • There exist no two integers m and n:
      • m and n being coprime
      • √ 2 =m/n
    √ 2 is irrational
    • select operator
    • (e.g. indirect proof)
    • There exist no two integers m and n:
      • m and n being coprime
      • √ 2 =m/n
    • mark statement with mouse
    • click „Pick“ button
  • 11. Group B: Bridge Building
    • Clear separation between:
      • Assumptions and Goals
      • Forward and Backward Reasoning
    Assumptions Forward Reasoning Goals Backward Reasoning
  • 12. Group B: Control Panel History System support
  • 13. Group B: Method Iconisation (Definition-) Expansion Contradiction Insert island (Definition-) Collapse
  • 14. Group B: Operator Application
  • 15. Bridge Construction Example Upper bank Lower bank √ 2 is not rational Action: definition application (Hypotheses) (Theorem)
    • There exist two integers m and n:
      • m and n being coprime
      • √ 2 =m/n
    Action: indirect proof Action: term rewriting
    • There exist no two integers m und n:
      • m and n being coprime
      • √ 2 =m/n
    m 2 =2*n 2 Contradiction
  • 16. Placing Islands
    • There exist two integers m and n:
      • m and n being coprime
      • √ 2 =m/n
    m 2 =2*n 2 m is even n is even
  • 17. Group C: Masking Operator Names Proof presented as trees of statements Edges = Story tellers „next do … to get … √ 2 is irrational √ 2 is rational  m  n: √ 2=m/n m, n are coprime
  • 18. Group C: Masking Operator Names √ 2 is irrational √ 2 is rational √ 2 is rational
    •  m  n: √2=m/n
    • m, n are coprime
    • √ 2 is irrational
    • contradtion
     m  n: √ 2=m/n m, n are coprime
  • 19. Group D: Notebooks
    • Linear proof style:
    • arrows denote relations
    • arrows labeled by operators
    √ 2 is irrational We assume: √ 2 is rational There exist two numbers n and m in Z, Being coprime, such that √2=n/m 2m=n 2 n 2 is even n is even
  • 20. Group D: Operator Application √ 2 is irrational We assume: √ 2 is rational There exist two numbers n and m in Z, Being coprime, such that √2=n/m 2m=n 2 n 2 is even n is even Search List all
  • 21. Conclusion
    • Used Argument: „As taught at school.“ (???)
      • A lot of „User Wants and User Needs“
      • Partly questionable
    • Discussion results:
    • Presentation
      • Simplified, nested statements
      • Bridge construction paradigm
      • Proof structuring (notebook, expansion, collapsing)
    • Standard interaction facilities
      • Copy&Paste, Drag&Drop, etc.
  • 22. Conclusion (2): System Support
    • Automation Support
      • Of simple steps
      • Verification of introduced islands
      • On demand completion of gaps
      • Copy&Paste for sub proofs
      • History
    • Feedback
      • Check proof/operator arguments
      • Help (e.g. explanations of operators)
    • Hints
      • General advice: „Derive a contradiction!“
      • Rank suggestions
      • Overcome failure (suggest suitable input arguments)
  • 23. Future Work
    • … towards a User Centered GUI ?
    • Prototype development
    • HCI evaluation