The learning manifold.
Every outcome is a location. Difficulty is curvature. Each learner is a position that moves. In production in schools across India.
The premise
Learning has a shape.
Most systems treat skills as a flat list: a learner has them or doesn't. Learning has structure. Concepts connect, skills depend on other skills, and a learner has a position in that structure. Activities reveal where they are. Time moves them through it.
The axes
Every subject has its own dimensions.
English isn't measured the same way as Maths. Maths isn't measured the same way as Science. The shape of what you're learning determines the shape of how it's measured. Each subject has its own coordinate system: the same structural framework, different axes.
Subject
English
Accuracy
getting to the right answer
Pronunciation
clear sound production
Fluency
smooth, well-paced delivery
Vocabulary
range and choice of words
Content
relevance and substance of ideas
Calibrated against CEFR — the Common European Framework of Reference.
Subject
Maths
Accuracy
getting to the right answer
Procedural Fluency
efficient, flexible execution
Conceptual Understanding
grasping why, not just how
Reasoning
justifying, generalising, pattern-spotting
Representation
moving between symbolic, visual, concrete forms
Calibrated against the TIMSS framework — content × cognitive domains.
Subject
Science
Accuracy
correct facts and applications
Inquiry
predicting, hypothesising, explaining
Conceptual Understanding
grasping underlying models
Representation
diagrams, graphs, equations
Vocabulary
domain-specific scientific language
Calibrated against the TIMSS Science Framework and NGSS practices.
Notice the universal axes. Accuracy appears in every subject: the universal “did you get to the right answer?” axis. Conceptual Understanding and Representation recur across Maths and Science. Each subject also has its own irreducible dimensions: Pronunciation belongs to English, Procedural Fluency to Maths, Inquiry to Science. The shape changes; the geometry does not.
The English and Maths geometries are built; Science has its dimension system defined, and its geometry is next.
The principle
Honest measurement, not invented numbers.
When an activity doesn't measure a dimension, we don't fake a score. We say nothing. A listening activity measures Accuracy, and only Accuracy. A speaking activity measures all five English dimensions. A reading-aloud activity measures Accuracy, Pronunciation and Fluency, but not Vocabulary or Content, because the child didn't choose the words.
A learner's position fills in as they cover more activity types. We never invent a score to fill a gap. That keeps the framework honest, maps cleanly to schools that don't run the full activity suite, and gives a natural progress story.
The calibration
Anchored to international standards.
The coordinate system is ours. The calibration anchors are public, international and standards-based.
For English, we calibrate against CEFR, the Council of Europe's six-level framework, used by language assessment bodies in over forty countries. CEFR's can-do descriptors define what a learner at A1, A2, B1, B2, C1 or C2 can actually do with the language. Our dimensions are the measurement axes; CEFR is where points on those axes correspond to recognisable proficiency levels.
For Maths, we calibrate against TIMSS, the IEA's international comparative study, which defines mathematical proficiency across content and cognitive domains. For Science, we draw on the TIMSS Science Framework and NGSS, both publicly published frameworks defining the practices and core ideas of scientific learning.
The cost of learning
Why ability gaps widen.
C = ρλ²
Cognitive cost equals complexity density times the square of how fast a given learner propagates understanding. Small differences in λ become large differences in cost exactly where the space is densest, which is why ability gaps widen at higher abstraction.
Curricula
One space, many charts.
A curriculum doesn't define the knowledge; it lays a set of coordinates over it. A child learning to add fractions in Cambridge Primary and a child learning the same thing in an Indian state-board classroom occupy the same region of the space. Adding a curriculum doesn't mean starting again. It means a new chart on a geometry that's already there.
Mapped so far
- Cambridge Primary English
- Cambridge Primary & Lower Secondary Mathematics
- CEFR
- UK National Curriculum
- NCERT / CBSE
- Tamil Nadu State Board
- Telangana State Board
- Montessori (ages 3–8)
Ages 3–8
Where it starts.
The early years have their own geometry. We built it from the Montessori method's own ordering of materials: how abstract the work is, how hard, how much of the hand and the senses it uses, and when it's left to the child. No “ahead”, no “behind”; every child at their own pace.
Maths
Solver first.
In maths, every question and every answer is computed by a solver before any language is generated. The language model explains; it never decides what's correct.
The trajectory
Learning is a path, not a level.
Most assessment systems give a learner a level. You're at A1. You're at Year 3 expected. A level is a snapshot. It tells you where the learner is, but not where they came from or where they're going.
We measure a learner's position continuously. That position is a vector, and it changes over time. The change is the learning. A teacher can see not just where a child is today, but where they were last week, which dimensions are growing fastest, and where the next step should take them.
It's in production. Our client Chrysalis uses it across its schools in India, and Outloud English, our own product, is built on it.
For researchers
We work with researchers.
The coordinate system produces an unusual kind of dataset: continuous, dimensional, cross-subject, and anchored to international standards. We work with academic researchers on learning trajectories, calibration and adaptive sequencing.