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.

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  • Designing a Proof GUI for Non-Experts

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

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