Semester Project: Mathematical Foundations Portfolio

Required Output Classification

Required output	Classification	Public/private guidance
Runnable project implementation and repository structure	Portfolio candidate	Polish the public repo only after tests pass, secrets are removed, and setup steps work from a clean checkout.
README with setup, inspection, verification instructions, and known limitations	Portfolio candidate	Make this public-facing if the project is safe to share; keep internal coursework notes in a private evidence folder.
Tests, traces, proofs, diagrams, benchmark outputs, or review notes required by the brief	Checkpoint evidence	Keep raw logs, benchmark runs, and reviewer comments private by default; publish summarized or reproducible versions when useful.
ADRs, design memos, runbooks, benchmark reports, and other high-effort engineering writeups	Portfolio candidate	These are worth polishing publicly when they tell a clear tradeoff story; otherwise keep them as private coursework evidence.
Final reflection, retrospective, or carry-forward notes	Checkpoint evidence	Keep candid self-assessment private unless rewritten as a concise public learning note.

Duration: Throughout semester, with final integration in Week 12
Deliverable: Comprehensive mathematical portfolio demonstrating mastery across all five modules

Project Overview

The Semester 1 project integrates proof techniques, combinatorics, probability, linear algebra, and problem-solving into a unified mathematical portfolio. Rather than five separate module projects, you will build interconnected components that demonstrate mathematical maturity and computational thinking.

Project Components

Component 1: Proof Construction Portfolio (Module 1 Integration)

Weight: 25% of project grade

Build a collection of 15 well-written mathematical proofs spanning different techniques and domains:

Required Proof Types:

• 3 direct proofs (including at least one function property proof)
• 2 contrapositive proofs
• 2 proof by contradiction
• 2 proof by cases
• 3 induction proofs (ordinary, strong, and structural)
• 2 set equality proofs using element-chasing
• 1 existence proof using combinatorial reasoning

Quality Standards:

• Each proof includes: clear statement, hypothesis identification, method justification, step-by-step reasoning, and conclusion
• Proofs are written for a peer audience (another Semester 1 student)
• At least 5 proofs must include geometric or computational intuition beyond pure symbolic manipulation
• Error analysis: document 3 common proof mistakes you made and how you corrected them

Integration Requirement: Connect proof techniques to later modules - use combinatorial counting in probability proofs, linear algebra reasoning in combinatorial arguments, etc.

Component 2: Combinatorial Analysis and Implementation (Module 2 Integration)

Weight: 25% of project grade

Develop both theoretical and computational combinatorial reasoning skills:

Theoretical Component:

• Solve 20 counting problems ranging from basic to advanced difficulty
• Include problems requiring: inclusion-exclusion, pigeonhole principle, generating functions, and graph theory
• Write detailed solution explanations showing method selection and verification
• Create 3 original counting problems with complete solutions

Computational Component:

• Implement combinatorial algorithms: permutation/combination generators, graph traversals (DFS, BFS), and Eulerian path detection
• Performance analysis: compare theoretical complexity with empirical measurements
• Visualization: create interactive demonstrations of 2 combinatorial concepts (e.g., graph coloring, tree generation)

Application Component:

• Choose one CS application area (algorithms, security, networking, or databases)
• Demonstrate how combinatorial reasoning applies to real problems in that domain
• Present findings in a 5-page technical report with implementation examples

Component 3: Probabilistic Reasoning and Simulation (Module 3 Integration)

Weight: 25% of project grade

Build both analytical and computational probability skills:

Analytical Component:

• Solve 15 probability problems covering: conditional probability, Bayes' theorem, expectation/variance, and common distributions
• Include real-world applications: algorithm analysis, system reliability, or statistical inference
• Demonstrate connection between combinatorial counting (Component 2) and probability calculations

Simulation Component:

• Implement Monte Carlo simulations for 3 probability problems where analytical solutions are complex
• Verify theoretical calculations by comparing with simulation results
• Include confidence intervals and statistical significance analysis
• Document simulation design choices and convergence behavior

Research Component:

• Investigate one probabilistic algorithm or data structure (e.g., Bloom filters, randomized quicksort, Monte Carlo methods)
• Analyze theoretical guarantees and empirical performance
• Present trade-offs between deterministic and probabilistic approaches

Component 4: Linear Algebra Applications Project (Module 4 Integration)

Weight: 15% of project grade

Choose ONE substantial application area and develop both theoretical understanding and practical implementation:

Option A: Computer Graphics

• Implement 3D transformation pipeline: modeling, viewing, and projection transformations
• Demonstrate geometric operations: rotations, scaling, translation using matrix operations
• Create interactive visualization showing how matrix operations affect 3D objects
• Analyze computational complexity of different transformation approaches

Option B: Machine Learning Foundations

• Implement linear regression using matrix operations (normal equations and gradient descent)
• Demonstrate principal component analysis (PCA) for dimensionality reduction
• Compare computational approaches and analyze numerical stability
• Apply techniques to real dataset with interpretation of results

Option C: Network Analysis

• Analyze graph properties using adjacency matrices and spectral graph theory
• Implement PageRank algorithm and analyze convergence properties
• Demonstrate relationship between graph structure and eigenvalue properties
• Compare different centrality measures and their computational requirements

Component 5: Meta-Cognitive Problem-Solving Analysis (Module 5 Integration)

Weight: 10% of project grade

Document and analyze your problem-solving development throughout the semester:

Process Documentation:

• Maintain weekly problem-solving journal documenting strategy selection and effectiveness
• Analyze 5 challenging problems where you got stuck: what strategies helped you break through?
• Compare your approach to problems at the beginning vs end of semester

Strategy Analysis:

• Create personal heuristic guide: when to use which problem-solving strategies based on problem characteristics
• Document 3 instances where you successfully transferred techniques between different mathematical domains
• Reflect on collaboration experiences: how did working with others improve your problem-solving?

Teaching Component:

• Create tutorial materials explaining one complex concept to a peer
• Document what worked and what didn't in your explanation approach
• Reflect on how teaching reinforced your own understanding

Integration Requirements

The portfolio must demonstrate connections across modules rather than treating them as separate subjects:

Required Cross-Module Connections:

1. Use combinatorial arguments in at least 2 probability proofs
2. Apply linear algebra techniques to at least 1 graph theory problem
3. Use probability reasoning to analyze at least 1 algorithm's expected performance
4. Apply proof techniques to verify correctness of at least 3 implementations
5. Document how problem-solving strategies from Module 5 improved your work in other modules

Timeline and Milestones

Week 3: Component Planning

• Choose linear algebra application area (Component 4)
• Set up project repository with documentation structure
• Complete first 5 proofs for proof portfolio
• Begin combinatorial problem collection

Week 6: Mid-Semester Check

• Complete 10 proofs across different techniques
• Finish theoretical component of combinatorial analysis
• Implement basic probability simulations
• Begin linear algebra application implementation

Week 9: Implementation Phase

• Complete combinatorial algorithm implementations
• Finish advanced probability analysis and simulation
• Complete linear algebra application with performance analysis
• Begin meta-cognitive analysis documentation

Week 12: Integration and Presentation

• Complete all 15 proofs with error analysis
• Finalize all implementations with documentation
• Complete cross-module integration analysis
• Prepare portfolio presentation (15 minutes)

Assessment Rubric

Excellent (90-100%)

• All components complete with high quality implementation and analysis
• Clear evidence of mathematical maturity and computational thinking
• Strong integration between modules with novel connections
• Exceptional problem-solving documentation and meta-cognitive analysis
• Professional-quality presentation and documentation

Proficient (80-89%)

• Most components complete with solid implementation
• Good mathematical reasoning with minor gaps
• Some integration between modules
• Adequate problem-solving analysis
• Clear presentation with good organization

Developing (70-79%)

• Basic requirements met but limited depth or integration
• Mathematical reasoning present but may have errors
• Limited connection between modules
• Minimal meta-cognitive analysis
• Acceptable but unpolished presentation

Incomplete (Below 70%)

• Missing major components or significant errors
• Insufficient mathematical reasoning or major misunderstandings
• No evidence of integration across modules
• No meaningful problem-solving analysis

Resources and Support

Technical Resources

• LaTeX: Use for mathematical typesetting and proof presentation
• Python/Jupyter: Recommended for implementations and visualizations
• Git: Version control for collaborative development and documentation
• Overleaf: Online LaTeX collaboration platform

Collaboration Policy

• Individual work required, but discussion and peer review encouraged
• You may work together on understanding concepts but must implement and write independently
• Cite any collaboration or assistance in your documentation
• Code may be shared for debugging but not copied wholesale

Getting Help

• Office hours: Mathematical concepts and proof verification
• Peer study groups: Problem-solving strategy discussion
• Online resources: Implementation techniques and computational methods
• Academic integrity: When in doubt, ask rather than assume

Submission Requirements

1. Digital Portfolio: Complete repository with all code, documentation, and analysis
2. Written Report: 20-page technical document integrating all components
3. Presentation: 15-minute portfolio overview with Q&A
4. Reflection Essay: 3-page analysis of mathematical growth and learning insights

Due Date: End of Week 12, with presentation during final exam period

This project demonstrates your readiness for Semester 2 by showing mathematical maturity, computational thinking, and the ability to integrate knowledge across domains - essential skills for advanced computer science coursework.

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Production-Style Project Brief

Use this project as a reviewable engineering brief, not only a completion exercise.

Problem statement

Write a one-paragraph statement covering the user, the problem, the constraint, and the outcome this project is meant to produce.

Required evidence

• working artifact or reproducible deliverable
• README with setup, inspection, and verification instructions
• tests, traces, proofs, diagrams, benchmark output, or review notes appropriate to the semester
• decision log with at least three meaningful tradeoffs
• known limitations section with explicit scope cuts

Review questions

1. What is the smallest vertical slice that proves the project works?
2. Which requirement is most likely to be misunderstood by a reviewer?
3. What did you deliberately not build, and why?
4. What evidence would convince someone else that the result is correct?

Done means

The project is done only when another technical reader can inspect the artifact, run or review the verification evidence, and understand the tradeoffs without a live explanation.

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Weekly Project Milestones

Use these milestones to keep the project from becoming a last-week scramble.

Milestone	Focus	Evidence
Start	Scope the smallest useful slice	problem statement, non-goals, first task list
Early build	Produce a walking version	runnable skeleton, first test, first committed artifact
Middle	Add the hard part	implementation note, trace/proof/benchmark/design decision
Review	Stress the weak point	failure case, debugging note, peer/self review, correction commit
Finish	Package for inspection	README, verification instructions, known limitations, reflection

Answer-Quality Examples

Quality	What it sounds like
Weak	"I built it because the module asked for it."
Acceptable	"It works for the required examples and I can explain the main idea."
Strong	"Here is the tradeoff I chose, the evidence that supports it, and the case where it would fail."
Portfolio-ready	"A reviewer can inspect the artifact, rerun the checks, and understand why this solution fits this semester's goals."

Future Capstone Connection

Before closing this project, write two sentences on how it could help the final capstone: one reusable technical skill and one artifact habit to preserve.

Calibration Materials

Use these learner-visible calibration materials before self-grading or requesting review:

• Answer key and solution outlines
• Rubric, passing standard, strong traits, and weak patterns