Technology
How the Engine Works
A geometric model of knowledge that tracks every student's position and determines the optimal next step.
The Core Idea
The Riemannian Knowledge Space
Every learning outcome in a curriculum is a point in a connected space. The distance between two points reflects how hard it is for a student to move from one concept to the next.
The metric tensor encodes this pedagogical cost — different curricula parameterise the same underlying space differently. A student's learning journey becomes a path through this space, and the engine's job is to find the optimal route.
C = ρλ²
Cognitive cost equals complexity density times the square of the semantic distance. This equation governs how the engine plans every learning path.
Adaptive Intelligence
How the Tutor Decides What to Teach
Four mathematical quantities, each mapping to a real tutoring decision. Every quantity earns its place by answering a concrete question about the student.
Entropy
“How uncertain are we about this student's understanding?”
Target the highest-uncertainty outcomes first — that's where assessment has the most value.
Divergence
“Is the student drifting from the optimal learning path?”
Redirect the conversation when the student's trajectory diverges from the geodesic.
Jacobian
“If the student masters this concept, what else improves?”
Prioritise outcomes with high propagation — mastering fractions unlocks decimals, percentages, ratios.
Green's function
“What's the influence radius of this learning event?”
Determine how far a single session's progress ripples through the knowledge graph.
Curriculum Mapping
Four Curricula. One Space.
A student learning “adding fractions” in Cambridge Primary occupies the same region of the knowledge space as a student learning the equivalent in the UK National Curriculum. The curriculum defines the coordinate system. The knowledge is the same.
Cambridge Primary
Cambridge Early Years
UK National Curriculum
Common Core
Proven at Scale
From production to product
This engine has run in production inside a school network of 100+ schools and 100,000+ students. ClassGrade Voice Tutor is the next evolution — bringing the same adaptive intelligence directly to families.
Meet ClassGrade →