把一条学习原理，连到你能改的那一处设计。

Systematic mathematical language and representation

Use explicit cumulative instruction to link precise mathematical language, quantities, selected visual or concrete models, and symbols.

Recommendation

Use a coherent representation system tied explicitly to language, symbols, and independent reasoning.

Mechanism

Stable mappings among language, quantity, diagrams, and notation reduce ambiguity and build both conceptual and procedural access.

Applicable contexts and prerequisites

Learner Stages

• primary
• secondary

Task Types

• arithmetic
• fractions
• word problems
• algebra readiness

Contexts

• core instruction
• targeted intervention

Lifespan Stages

• primary
• middle

• Diagnosis of the current mathematical bottleneck
• A coherent limited representation set
• Modeled and guided practice

Constraints and risks

• Explain how each representation maps to the abstraction
• Fade models when independent reasoning is ready

• Too many competing models
• Manipulatives becoming procedures without meaning
• Language demands hiding mathematical knowledge

Observable evidence

• Indicator

  Conceptual and procedural mapping across forms

  Measure

  Explanation, translation among representations, procedural accuracy, and generalization

  Timing

  During intervention and on a novel representation

  Caveat

  Successful manipulation without explaining the mathematical relation is insufficient

Evidence

strong

IES/WWC rates systematic instruction, mathematical language, and selected representations strongly for elementary intervention.

Confidence

high

Directness

direct-intervention

Generalizability

Elementary learners struggling with mathematics; adapt model and language to content.