Number sense training: subitizing, relational thinking (RFT/Posner), and relational integration across hierarchical levels.
Eucalculia is an evidence-based training tool designed to strengthen number sense through three complementary mechanisms: subitizing (instant quantity recognition), relational evaluation (Posner paradigm applied to number properties), and relational integration (computing and comparing derived numerical relations across hierarchical levels).
The relational integration system (Levels 1 and 2) operationalizes Halford's relational complexity hierarchy in the numerical domain. Level 0 trains binary relational evaluation; Level 1 adds computation on retained values; Level 2 requires comparing across computations — the same cognitive demand that predicts fluid intelligence.
Two quantities flash side by side, then disappear. A relational question appears: "Same parity?", "Sum > 10?", etc. You answer YES or NO. You must encode both quantities as abstract numerical representations because the question is hidden during presentation.
After two Level 0 trials you are asked a numeric question that combines both pairs — e.g. "Sum of all four values?" or "Difference of the two maximums?". There is no hint: both pairs must be retained in working memory as abstract numbers. Enter the answer on the numpad.
This is the critical step: Level 0 only requires evaluating a relation; Level 1 requires retaining values and computing new ones — the transition from relational evaluation to relational integration.
After two Level 1 trials, a Meta² question compares the numerical outputs you computed. "Was Calc₁ > Calc₂?", "Same parity?", "Differ by more than 3?" — using the same relational vocabulary as Level 0, but applied to numbers you constructed mentally, never saw on screen.
After two Level 1 trials you are asked a boolean question that compares the two numeric answers you computed — e.g. "Was the first answer larger?" or "Same parity?". You must have retained both L1 answers in memory to respond. This is a ternary relation (a relation between two relations).
After two Level 2 trials, a Meta³ question compares the two correct YES/NO answers from the previous Meta² questions. The wording is explicit — for example: "Were both previous Meta² answers YES?" or "Was the first previous Meta² answer NO?" You must retain both prior booleans in working memory despite all the numerical/relational processing between them. This is Halford's quaternary level — the theoretical ceiling of human relational processing.
Ground truth is the correct answer to each L2, not what you responded. If you got an L2 wrong, the REVIEW feedback tells you the correct value; use it to update your retained boolean before L3 arrives.
Trials follow nested cycles: L0 → L0 → L1 repeats twice to set up an L2, and two L2s set up an L3. At Meta³ enabled, the full cycle is 15 trials. L1 always spans the two preceding L0 pairs, L2 the two preceding L1 answers, L3 the two preceding L2 booleans. Temporal proximity ensures integration — not episodic recall — is the cognitive bottleneck.
For difference operations, the question wording indicates the direction (e.g. "first pair's total minus second pair's total"). The system selects the direction that yields a non-negative answer, but you still have to compute which way the question is pointing.
L2 thresholds are calibrated against an exact analytical distribution for the actual pair of L1 operations that produced r₁ and r₂, under the modelled value range [1..m]. Because the live L0 generator also has category weights, filters and adaptive history, this is an exact model-level calibration rather than a claim that every effective session distribution is mathematically uniform. When adjacent integer thresholds straddle the target rate, the system samples between them across trials.
L3 ground-truth is the correct answer to each prior L2, not what you responded. If you got an L2 wrong, use the REVIEW feedback to update your retained booleans before L3 arrives. This trains both quaternary relational integration (Halford's theoretical ceiling) and feedback-driven belief updating.
Subitizing mode: each number has its own exposure time (ET). Correct fast responses shorten ET; errors increase it. Less-mastered numbers appear more frequently. Level-up when all values at the current range are mastered.
Relational mode: a single global ET applies to all stimuli — individual values are not the unit of learning, relations are. Level-up requires a rolling window of L0 trials at the current level with ≥85% accuracy and average correct RT ≤1200 ms (window size scales with level difficulty).
Level 0 categories are filtered by whether the current value range supports them (e.g. "Sum > 10?" only appears once the range is large enough). Each L0 trial is an independent Bernoulli(0.5) over YES/NO: the generator draws a target outcome first, then builds a pair that satisfies it — so YES and NO are structurally balanced, not statistically hoped-for.
L0 categories are weighted by per-category accuracy and RT (harder categories appear more). L1 operations are weighted by accuracy. L2 picks category uniformly, balances target YES/NO per trial, computes thresholds from analytical operation-pair distributions, avoids degenerate zero-threshold prompts, and logs YES rate, accuracy, RT and fallback rate by category. L3 also picks category uniformly and balances target YES/NO per trial; its first/second-answer category includes explicit YES and NO variants so natural-rate fallbacks should be exceptional and measurable.
Developed by Alberto Flaño Lombardo
linkedin.com/in/alberto-flaño-lombardo-762618259Based on research in numerical cognition, subitizing, Relational Frame Theory, the Posner paradigm, and Halford's relational complexity theory.