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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 GUIfor Non-Experts
Evaluation of an Experiment
Martin Homik, Andreas Meier
Presentation by Christoph Benzmüller
UITP 2005, Edinburgh
ActiveMath GroupGerman Research Center for Artificial Intelligence
DFKI GmbH, Saarbrücken
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!
Motivation (2)
MIPPA Project goals:• Interactive learning tool for math. proof• Underlying proof engine: Proof planner MULTI
Target group:• Undergraduate students• A-level pupils
Expert GUI: Loui
Towards a User Centered GUI
First step:
• Paper&Pencil student experiment
• Primary task:– Observe basic user wants and needs
4 Groups• 2 students in eachBackground:• Computer Science, Math, LogicNo design restrictions:• creativity/underlying system• use/invent functionalities freely
Experiment Setting
Design (120 min) Presentation (15 + 10 min) Discussion
Example Theorem: Irrationality of √2Use:• Definitions• Term rewriting• Island introduction• Contradiction
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
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*m2=n2 (1), which results in the fact that n2 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*m2=4*k2 which can be simplified to m2=2*k2. Hence, m2 and m are even as well. This is a contradiction to the fact that n,m are supposed to have no common divisor.“
Group A: Text-based
Textual presentation of a proof.The same way as taught at school.
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
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)
• 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
• mark statement with mouse• click „Pick“ button
Group B: Bridge Building
AssumptionsAssumptions
Forward ReasoningForward Reasoning
GoalsGoals
Backward ReasoningBackward Reasoning
Clear separation between:• Assumptions and Goals• Forward and Backward Reasoning
Clear separation between:• Assumptions and Goals• Forward and Backward Reasoning
Group B: Control Panel
HistoryHistory
System supportSystem support
Group B: Method Iconisation
(Definition-) Expansion
ContradictionInsert island
(Definition-) Collapse
Group B: Operator Application
Bridge Construction ExampleUpper bank
Lower bank
√2 is not rational
Action: definition application
There exist no two integers m und n:• m and n being coprime • √2 =m/n
(Hypotheses)
(Theorem)
There exist two integers m and n:• m and n being coprime• √2 =m/n
m2 =2*n2
ContradictionAction: indirect proof
Action: term rewriting
Placing Islands
There exist two integers m and n:• m and n being coprime• √2 =m/n
m2 =2*n2
m is even n is even
Group C: Masking Operator Names
√2 is irrational
√2 is rational
mn: √2=m/n m, n are coprime
Proof presented as trees of statementsEdges = Story tellers „next do … to get …
Proof presented as trees of statementsEdges = Story tellers „next do … to get …
Group C: Masking Operator Names
√2 is irrational
√2 is rational√2 is rational
• mn: √2=m/n• m, n are coprime
• √2 is irrational• contradtion
mn: √2=m/n m, n are coprime
Group D: Notebooks
√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=n2
n2 is even
n is even
Linear proof style:• arrows denote relations• arrows labeled by operators
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=n2
n2 is even
n is even
Group D: Operator Application
Search Search
List allList all
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.
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)
Future Work
… towards a User Centered GUI ?
• Prototype development
• HCI evaluation