Making Mathematical Reasoning Fun

Jason Hallstrom (Clemson), Joan Krone (Denison), Joseph E. Hollingsworth (IU Southeast), and Murali Sitaraman(Clemson)

This workshop is funded in part by NSF grant

DUE-1022941

GoalsReasoning Across the Curriculum

Not just in Discrete MathFundamental part of CSMotivating example: binary search “proven” correct

Supporting ToolsSupporting MethodsApplicable to both large universities and small

colleges

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PartnersAlabamaClemsonCleveland StateDenisonDepauwIU SoutheastRamapo CollegeVirginia Tech NVCWestern Carolina

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What reasoning skills are necessary?Concept InventoryBoolean Logic Standard Logic Symbols, Standard

Proof Techniques

Discrete Math Structures Sets, Strings, Numbers, Relations, and other mathematical theories as needed

Precise Specifications Mathematical Descriptions of Software interfaces for clients and implementers.Math models for structuresPre and Post conditions for operations.

Modular Reasoning Each Module needs to be proven correct only once.

Verification Conditions Mathematical Assertions equivalent to the correctness of the program.

Correctness Proofs Application of Proof Techniques to the program

Why?Current software is too large for one person to

understand.Students need tools for dealing with all sizes of

projects.Maintenance makes up the majority of jobs.Students need to separate specifications from

implementations.

Courses at All LevelsBeginning level: CS110 – Intro to Programming,

CS174 – Discrete math at Denison, (others – CPSC101 – CSI, CPSC102 – CS2, Discrete Math at Clemson)Use of collaborative approachUse of specifications Reasoning assistant tool

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Intermediate level: CPSC215 – Software Foundations, (others: CPSC212 – Data Structures) at ClemsonContract specifications – comparing informal specs

with formal specsMathematical modeling – abstractionGenerating test data from specsReasoning assistant tool

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Advanced level: CS373 – Theory of Programming Languages and CS349 – Software Engineering at Denison, CP372 – Software Engineering at ClemsonFormal specificationsProofsVC generator toolContract based team development using RESOLVE

compiler

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All LevelsCollaborative Approach

Pairs or small groupsIn class or homework

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Collaborative MethodPairs or small groupsWith or without toolsEach team presents their findingsCollaboration both within teams and among teams

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Selective AdaptationPick and choose appropriate reasoning concepts

and/or toolsFaculty expertiseStudent background

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One Example: Software Engineering CourseUsual Topics

Requirements analysisDesign and specificationComponent-based implementationQuality assurance

Formal Reasoning

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ObjectivesRead formal specifications

Create test points from the specsUse component specifications to build larger

systemsWork in teams

Carry out formal verification of componentsUse automated rules

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MethodsCollaborative learning

Teams of 2 to 4 membersRead specsImplement specsVerify implementationsBuild larger systems.

Using the Toolshttp://www.cs.clemson.edu/group/resolve

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SummaryImportance of Reasoning across the CurriculumTools to Support ReasoningCollaborative Pedagogy includes collaboration

between students and between students and faculty

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Some WorkBinary search specificationsJavaC++Any other languageAre the algorithms correct?Do the implementations work?What’s the difference?