Learning Graph Quality Metrics Report
Overview
- Total Concepts: 200
- Foundational Concepts (no dependencies): 3
- Concepts with Dependencies: 197
- Average Dependencies per Concept: 1.41
Graph Structure Validation
- Valid DAG Structure: ❌ No
- Self-Dependencies: None detected ✅
- Cycles Detected: 0
Foundational Concepts
These concepts have no prerequisites:
- 1: Number
- 2: Variable
- 3: Constant
Dependency Chain Analysis
- Maximum Dependency Chain Length: 16
Longest Learning Path:
- Variable (ID: 2)
- Coefficient (ID: 4)
- Term (ID: 5)
- Monomial (ID: 42)
- Binomial (ID: 43)
- Trinomial (ID: 44)
- Polynomial (ID: 45)
- Multiplying Polynomials (ID: 51)
- Special Products (ID: 53)
- Perfect Square Trinomial (ID: 55)
- Completing the Square (ID: 173)
- Quadratic Formula (ID: 174)
- Discriminant (ID: 175)
- Nature of Roots (ID: 176)
- Complex Numbers (ID: 177)
- Imaginary Unit (ID: 178)
Orphaned Nodes Analysis
- Total Orphaned Nodes: 103
Concepts that are not prerequisites for any other concept:
- 10: Commutative Property
- 11: Associative Property
- 13: Identity Property
- 17: Real Numbers
- 25: Rational Exponents
- 26: Negative Exponents
- 27: Zero Exponent
- 29: Quotient Rule for Exponents
- 30: Power Rule for Exponents
- 31: Scientific Notation
- 35: Simplifying Radicals
- 40: Simplifying Expressions
- 41: Expanding Expressions
- 47: Leading Coefficient
- 49: Adding Polynomials
- 50: Subtracting Polynomials
- 52: FOIL Method
- 61: Factoring Perfect Squares
- 62: Factoring by Grouping
- 63: Prime Factorization
...and 83 more
Connected Components
- Number of Connected Components: 1
✅ All concepts are connected in a single graph.
Indegree Analysis
Top 10 concepts that are prerequisites for the most other concepts:
| Rank | Concept ID | Concept Label | Indegree |
|---|---|---|---|
| 1 | 102 | Function | 16 |
| 2 | 1 | Number | 13 |
| 3 | 45 | Polynomial | 11 |
| 4 | 132 | Slope | 10 |
| 5 | 56 | Factoring | 9 |
| 6 | 2 | Variable | 8 |
| 7 | 21 | Exponent | 8 |
| 8 | 129 | Graph of a Line | 8 |
| 9 | 146 | System of Equations | 8 |
| 10 | 71 | Solving Equations | 7 |
Outdegree Distribution
| Dependencies | Number of Concepts |
|---|---|
| 0 | 3 |
| 1 | 121 |
| 2 | 72 |
| 3 | 4 |
Recommendations
- ⚠️ Many orphaned nodes (103): Consider if these should be prerequisites for advanced concepts
- ℹ️ Long dependency chains (16): Ensure students can follow extended learning paths
- ℹ️ Consider adding cross-dependencies: More connections could create richer learning pathways
Report generated by learning-graph-reports/analyze_graph.py