The Formal Operational schemata used in this dialogue

Schema

Amira (2A Early Concrete)

Ben (2B Late Concrete)

Chloe (3A Early Formal)

Daniela (3B Late Formal)

Proportionality & Inverse Proportionality

✖ Only recognises direction (‘hotter → faster’)

✓ Simple direct link (‘double strength → twice as fast’)

✓✓ Interprets slope and relates rate to concentration

✓✓✓ Expresses proportional laws (rate ∝ [A], rate ∝ [A][B])

Classification

✓ Perceptual (hot/cold, fast/slow)

✓ Functional (weak/strong acids)

✓✓ Differentiates variable types (temp, conc., surface area)

✓✓✓ Uses scientific variable control (independent/dependent)

Compound Variables

✖ Considers one factor at a time

✖ Treats each variable separately

✓ Identifies multiple variables but not their interaction

✓✓✓ Integrates multiple variables (compound proportionality)

Probability

✖ None; deterministic reasoning

(✓) Implies frequency, not probability

✓ Recognises some collisions are successful

✓✓✓ Explains activation energy and probability of effective collisions

Correlation

✓ Qualitative (hotter → faster)

✓ Quantitative rule (double → double)

✓✓ Uses graphs, links slope to rate

✓✓✓ Quantifies relationships; relates data correlation to rate law

Formal Models

✖ None

✖ None beyond rule statements

✓ Graphical model (concentration–time graph)

✓✓✓ Algebraic and conceptual models (rate = k[A]^m[B]^n)

Equilibrium

✖ No awareness of reversibility

✖ Views reactions as one-way

(✓) Notes rate decreases as reactants used

✓ Emerging concept; mentions half-life, decay, dynamic balance