Mathematical Learning, Instruction, and Assessment
What to know
This domain applies learning theory and assessment practice. Know Piaget's stages and the move from concrete to formal reasoning; Vygotsky's zone of proximal development and scaffolding; Bruner's concrete-representational-abstract sequence and spiral curriculum; and the role of prior knowledge and misconceptions. Instruction items ask about task selection, questioning, discourse, differentiation, manipulatives, technology, and support for English learners and students with disabilities. Assessment items distinguish formative from summative purposes and ask how to use data to adjust instruction.
Traps
Do not label every quiz 'formative'-formative assessment is defined by how results are used to adjust teaching, not by the format. Hands-on materials do not guarantee conceptual understanding; the discussion and recording matter. Scaffolds should be gradually removed, not permanent. Modifications change what a student is expected to learn; accommodations change how. A common distractor overgeneralizes one theory or claims a single strategy works for all learners.
Worked example
After a lesson on equivalent fractions, a teacher gives a three-item exit ticket. Most students miss the item requiring 2/3 = ?/12. The teacher reteaches with fraction strips the next day and groups students who are ready for extension. Rule of thumb: collect evidence, sort it into reteach, ready, and extension, and let the data drive the next day's plan.