develops skill

Purpose

This edge represents that the learning module develops (teaches) the skill as a learning outcome. The develops-skill relationship connects modules to the skills students gain by completing them.

Relationship Type

Edge Type: develops-skill (learning outcome)

Direction: learning-module → skill (module develops skill)

Participants:

• Source: v:learning-module:<module-name> (the module that teaches)
• Target: v:skill:<skill-name> (the skill that is developed)

Meaning

This edge indicates that <module-name> teaches <skill-name> as a learning outcome. Students who complete this module gain this skill, transitioning to a new learning state with an expanded skill set.

This represents the output signature of the module:

• Input: requires-skill edges (prerequisites)
• Output: develops-skill edges (learning outcomes)

Skill Accumulation Pattern

This edge enables supermodular skill accumulation:

student-after.skills = student-before.skills ∪ module.develops-skills

Where module.develops-skills = {all skills with develops-skill edges from module}

The skill set grows (supermodular) through module completion, and the cardinality is monotonically increasing.

Examples

Foundational Module (Develops First Skills):

# Module develops foundational skills from zero prerequisites
source: v:learning-module:simplicial-complex-fundamentals
target: v:skill:simplicial-complex-fundamentals
# Students gain this skill by completing the module

Intermediate Module (Develops Advanced Skills):

# Module develops new skills building on prerequisites
source: v:learning-module:verification-fundamentals
target: v:skill:verification-patterns
# Students who already have prerequisites gain this skill

Module with Multiple Outcomes:

# Multiple develops-skill edges for multiple learning outcomes
source: v:learning-module:advanced-topology
target: v:skill:homology-theory

source: v:learning-module:advanced-topology
target: v:skill:topological-analysis
# Students gain ALL skills the module develops

Participation in Faces

develops-skill edges participate in completion faces:

type: face/completion
vertices:
  - v:student:<student-before>        # initial learning state
  - v:learning-module:<module-name>   # develops-skill edge source
  - v:student:<student-after>         # new learning state

The completion face represents:

• student-before completes module
• module develops skills (this edge)
• student-after has student-before.skills ∪ module.develops-skills

Module Signature Pattern

Complete Module Signature:

Module: v:learning-module:example

Input (Prerequisites):
  requires-skill → v:skill:prerequisite-A
  requires-skill → v:skill:prerequisite-B

Output (Learning Outcomes):
  develops-skill → v:skill:outcome-X
  develops-skill → v:skill:outcome-Y

Signature: (prerequisite-A, prerequisite-B) → (outcome-X, outcome-Y)

Students must have input skills, gain output skills.

Constraints

• Source MUST be a learning-module vertex (v:learning-module:*)
• Target MUST be a skill vertex (v:skill:*)
• Each module-skill pair should have at most one develops-skill edge
• Skill must be defined (skill vertex should exist)
• Module must be defined (learning-module vertex should exist)
• Developed skill should NOT also be a required skill for the same module (module doesn't teach what it requires)

Anti-Pattern: Circular Development

Invalid:

Module A develops Skill X (needed for Module B)
Module B develops Skill Y (needed for Module A)
Both modules require each other's outputs as inputs
❌ CIRCULAR DEPENDENCY

Valid:

Module A (foundational) develops Skill X
Module B requires Skill X, develops Skill Y
Module C requires Skill Y, develops Skill Z
✅ Valid linear progression

Skill Set Computation

Given a completion face (student-before, module, student-after):

# Compute student-after skills
student_after_skills = (
    student_before_skills
    | module_develops_skills  # Set union (supermodular)
)

# Cardinality is monotonically increasing
assert len(student_after_skills) >= len(student_before_skills)

Verification

This edge can be verified using:

python scripts/verify_template_based.py 01_edges/<edge-file>.md --templates templates

Note: This template defines the develops-skill relationship representing module learning outcomes. It is essential for computing student skill set transitions and forms the output signature of learning modules. Together with requires-skill (input signature), it defines the complete module transformation: input-skills → module → output-skills.