Arithmetic to Algebra Framework

Below is an outline of the cognitive learning trajectories for a framework that represents mathematics from arithmetic to algebra.

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Cognitive Learning Trajectories

  1. Counting & Cardinality → subitizing, one-to-one, last-number-as-total.
  2. Unitizing & Place Value → compose/decompose tens and hundreds; number line as measure.
  3. Additive Reasoning → comparison vs. combination; make-ten, compensation; inverse (addition ↔ subtraction).
  4. Multiplicative Reasoning → equal groups, arrays/area, scaling; properties highlighted via arrays (commutative) and area/partial products (distributive); inverse (multiplication ↔ division).
  5. Fraction Reasoning → units as parts of a whole and as measures; equivalence; fraction × whole as operator; benchmark reasoning on the number line.
  6. Ratio/Rate & Proportional Reasoning → covariation (how two things change together), “per” language, double number lines/tables, constant of proportionality and slope.
  7. Signed Numbers & Generalization → extend number line through zero; relational thinking with unknowns; express patterns with variables; structure and form (e.g., a(b+c)).

Crosscutting strands: properties (commutative, associative, distributive; identities and inverses), equality as balance, representation fluency (concrete → pictorial → abstract).