Differentiated Instruction: Readiness Gap Simulator

Classroom Readiness

Readiness spread 20

Instructional pace 1.0×

Class average mastery

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Outcome spread (σ)

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Students in productive-challenge zone

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How it works

Every real class walks in with a spread of prior knowledge, not a single starting line. One-size-fits-all instruction has to aim somewhere — usually the class average — which means it's pitched too hard for students below that line and too easy for students above it. Learning is fastest in a narrow "productive challenge" zone: material pitched just beyond what a student can already do (their zone of proximal development), not miles beyond it and not already mastered.

gap_i = L_i − R_i(t)
efficiency_i = exp(−(gap_i − gap*)² / 2σ_g²)
dR_i/dt = pace · efficiency_i

Uniform mode — every student is taught at the same fixed level L, set at the class's starting average. Students near that average learn well; students far below get a gap too large to bridge (frustration, disengagement) and students far above get a gap too small to matter (boredom).
Differentiated mode — each student's instructional level L_i continuously tracks their own current mastery plus the same optimal challenge gap, so every bar in the room sits in its own productive-challenge zone at once.
Readiness spread — how varied the class's starting knowledge is; a wider spread makes uniform instruction fail more students while barely touching the differentiated run.
Instructional pace — how much content moves per week; it scales growth but doesn't change who's in their challenge zone.

Watch the outcome spread (σ): under uniform instruction it tends to widen as the class runs — the already-ahead pull further ahead and the behind fall further behind. Under differentiated instruction the whole distribution rises together, because nobody is stuck too far outside their own zone.

Uniform instruction

Low mastery Mid mastery High mastery

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