What is a cognitive graph? How Plan2Skill measures learning differently

What most learning apps measure — and why it's the wrong thing

Open any language learning app and you'll see your progress measured in a familiar way: words learned, lessons completed, streaks maintained. These are counts of nodes — discrete items you've touched.

The logic seems reasonable. More words memorised means more progress, right? But anyone who has been through a language course — or spent 200 days on a streak app only to freeze when a native speaker speaks naturally — has felt the gap between node-counting and actual fluency.

Plan2Skill measures something different: not what you know, but how your knowledge connects. We count edges.

Nodes, edges, and the cognitive graph

A cognitive graph is a model of your knowledge as a network. Each concept you know is a node. Each connection between concepts — the relationship, rule, contrast, or pattern linking them — is an edge.

Here is a concrete example using Spanish irregular verbs:

The node is easy to look up. The edge — the rule linking tener, poner, valer, and salir — is what you need to produce Spanish under pressure. Fluency lives in the edges.

This principle extends well beyond languages. In physics, knowing F = ma is a node. Knowing when to apply Newton's second law instead of conservation of momentum — and why each approach works in different problem types — is an edge. In programming, knowing what a LEFT JOIN does is a node. Knowing when a LEFT JOIN beats an INNER JOIN for a specific data shape, and why, is an edge.

The five theorists behind CTLT

Plan2Skill's approach is grounded in what we call Cognitive Topological Learning Theory (CTLT) — a synthesis of five decades of cognitive science research. The name is ours. The science belongs to five researchers who independently identified the same truth: learning is building connections, not accumulating facts.

Lev Vygotsky — Zone of Proximal Development (1934). Vygotsky showed that the most productive learning happens in the gap between what you can do alone and what you can do with guidance. In graph terms: the edges adjacent to your existing strong connections, but not yet mapped. Plan2Skill targets exactly this frontier.

Donald Hebb — Co-activation rule (1949). "Neurons that fire together, wire together." When two concepts are activated together repeatedly, the connection between them strengthens. Plan2Skill exercises edges — not just nodes — so the connections, not just the facts, get strengthened.

Collins & Loftus — Spreading activation (1975). Their model showed how retrieving one concept automatically activates related concepts through a network. A strong cognitive graph means that activating one node spreads quickly and accurately to connected nodes. Plan2Skill builds the structure that supports that spread.

Robert Bjork — Desirable difficulty (1994). Bjork demonstrated that making retrieval slightly harder — spacing practice, varying conditions, testing rather than re-reading — produces much stronger long-term retention. Plan2Skill calibrates quest difficulty to sit precisely at the edge of your current capability, making each retrieval maximally productive.

Manu Kapur — Productive failure (2008). Kapur's research showed that struggling with a problem before receiving instruction leads to deeper learning than instruction-first approaches. Encountering difficulty activates prior knowledge and creates a "prepared mind" for the incoming information. Plan2Skill uses this principle in quest sequencing: you encounter the edges before they're fully formed.

What your graph looks like in Plan2Skill

When you complete quests in Plan2Skill, the platform builds a personal cognitive graph. Every question that tests a relationship between two concepts — not just whether you know a fact, but whether you know how it connects to something else — adds or strengthens an edge in your graph.

The graph is used to decide what to test next. The algorithm looks for:

• Weak edges — connections you've formed but that snap under pressure. These get retested.
• Frontier edges — connections adjacent to strong ones, ready to be formed. These become the next quest.
• Gap edges — connections you haven't attempted yet but that are structurally important to your goals. These become future quests.

The result is a learning path that isn't generic — it's shaped by the exact topology of your current understanding.

Why edges matter more than nodes

A learner with 500 nodes and 200 edges has memorised a lot of isolated facts. They may score well on multiple-choice tests, but they'll struggle with production, transfer, and application under pressure.

A learner with 200 nodes and 600 edges has a dense, well-connected graph. They can generate answers, apply rules to novel contexts, and reason across domains. They have, in the everyday sense of the word, understood the subject.

Traditional learning measurement — streaks, completion rates, vocabulary counts — optimises for the first profile. Plan2Skill optimises for the second.

"Fluency, in any domain, lives in the edges — not the nodes."

A note on where we are

Plan2Skill is in alpha — approximately 50 learners, Q2 2026, currently serving the Languages domain. The cognitive graph architecture is live. We don't yet have peer-reviewed efficacy data; we're building toward a pre-print targeted for Q4 2026 peer-review submission.

We publish our metrics monthly at /metrics so you can watch the evidence grow in public, not in a press release.

If you want to go deeper into the science, the full science page covers CTLT in detail — the five theorists, the formulas we don't publish (our alpha IP), and the specific claims we do and don't make.

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