NBPTS AYA Math cheat sheet

Van Hiele levels: visualization, analysis, informal deduction, formal deduction, rigor-students progress sequentially, not by age alone.

Common misconceptions: (a+b)^2 = a^2 + b^2; multiplying always makes larger; dividing always makes smaller; negative exponents yield negative results.

Piagetian stages: concrete operational (ages 7–11) supports conservation and reversibility; formal operational (11+) supports hypothetical reasoning.

Conceptual knowledge = understanding relationships; procedural knowledge = knowing how to execute steps; both are necessary for fluency.

Representational fluency: move among concrete, pictorial, symbolic, verbal, and contextual representations of the same idea.

Cognitive load theory: limit extraneous load; chunk new material; provide worked examples for novices, faded guidance for experts.

Mathematical knowledge for teaching (MKT) includes subject matter knowledge and pedagogical content knowledge (Shulman).

Developmental trajectory example: counting → additive reasoning → multiplicative reasoning → proportional reasoning → algebraic reasoning.

Language demands: distinguish 'less than' (subtraction) from 'less than' (comparison); 'of' often signals multiplication in fraction contexts.

Backward design: identify desired results (standards), determine acceptable evidence (assessment), then plan learning experiences.

Task selection: use high-cognitive-demand tasks (Stein et al.)-doing mathematics, procedures with connections-not just memorization.

Three-phase lesson structure: launch (activate prior knowledge, set purpose), explore (student work), summarize (synthesize and formalize).

Universal Design for Learning (UDL): multiple means of representation, action/expression, and engagement.

Learning trajectory: goal, developmental progression, and instructional tasks matched to each level.

Formative assessment plan: pre-assessment, hinge questions, exit tickets, and mid-lesson checks aligned to objectives.

Pacing: allocate ~50% of time to sense-making and discussion, ~30% to guided practice, ~20% to independent application.

Differentiation: tier tasks by complexity, not volume; use parallel tasks with same big idea at different entry points.

Standards alignment: unpack standards into learning targets that specify observable student actions (e.g., 'explain why', 'model with').

Talk moves (Chapin, O'Connor, Anderson): revoicing, restating, reasoning, adding on, wait time, turn and talk.

Five practices for orchestrating discussion (Smith & Stein): anticipating, monitoring, selecting, sequencing, connecting.

Wait time: 3–5 seconds after a question increases response length and student-to-student interaction.

Accountable talk: students must justify claims with evidence and respond to peers' reasoning.

Questioning progression: funneling (teacher seeks answer) vs. focusing (teacher probes student thinking).

Productive struggle: provide tasks that require effort but are within reach; avoid rescuing too quickly.

Discourse norms: listen to understand, critique ideas not people, explain and justify, build on others' thinking.

Representation sequencing: start with concrete/pictorial, bridge to symbolic, then to abstract generalization.

Formative feedback: specific, actionable, and timely; focus on the process and strategies, not just correctness.

Validity: assessment measures what it claims; reliability: consistency of results across occasions or raters.

Formative assessment cycle: elicit evidence → interpret → act (feedback, adjustment) → re-engage students.

Rubric design: analytic rubrics break performance into dimensions (e.g., conceptual understanding, procedural fluency, communication).

Item types: selected response, constructed response, performance task; match item type to learning target.

Hinge question: a single diagnostic question used mid-lesson to decide whether to move on or reteach.

Error analysis: classify errors as conceptual, procedural, or careless; each calls for different instructional response.

Standard setting: cut scores should be based on expected performance, not norm-referenced ranking.

Data use: triangulate multiple sources (observations, work samples, quizzes) before making instructional decisions.

Feedback timing: immediate for procedural skills, delayed for complex problem solving to allow reflection.

Equity vs. equality: equity provides differentiated support so all students can reach high expectations.

Culturally responsive teaching: connect mathematics to students' lived experiences and community contexts.

Stereotype threat: reduce identity-based anxiety by emphasizing belonging and growth mindset.

Accessible materials: provide multiple representations, language supports, and assistive technology as needed.

Classroom norms: mistakes are valued as learning opportunities; all students are expected to participate and explain.

Grouping: avoid fixed ability grouping; use flexible, heterogeneous groups with rich tasks.

Participation structures: random calling, think-pair-share, and small-group roles increase equitable engagement.

Safe environment: physical layout allows collaboration; routines are predictable; respect is modeled.

Language supports: sentence frames, word walls, and bilingual glossaries for English learners.

Reflection cycle: describe → analyze → interpret → plan action (e.g., Gibbs or Kolb).

Lesson study: plan collaboratively, observe a research lesson, debrief, revise, and teach again.

Video analysis: review classroom recordings to identify patterns in questioning and discourse.

Professional learning communities: focus on student data, shared goals, and collective inquiry.

Self-assessment tools: teaching journals, student feedback, and peer observation protocols.

Action research: identify a problem, implement an intervention, collect data, and adjust practice.

Growth mindset for teachers: view challenges as learning opportunities; seek feedback and iterate.

Reflective questions: What worked? What evidence do I have? What will I change next time?

Continuing education: attend conferences, read research, participate in online communities, pursue advanced certification.

Family engagement: communicate regularly, offer multiple modes (email, phone, app), and invite input.

Cultural humility: learn about families' backgrounds and avoid deficit assumptions.

Homework policy: clarify purpose, provide examples, and avoid requiring parent teaching of new content.

IEP/504 collaboration: coordinate accommodations with special education staff and families.

Collegial collaboration: co-planning, co-teaching, and peer observation improve instruction.

Communication norms: use plain language, avoid jargon, and provide translated materials when possible.

Conflict resolution: listen actively, focus on student needs, and seek mutually agreeable solutions.

Community resources: connect families to tutoring, libraries, and after-school programs.

Reporting progress: use standards-based descriptions and specific examples of student work.

Modeling cycle: identify problem → formulate model → compute → interpret → validate → report.

Problem-solving heuristics (Pólya): understand, plan, carry out, look back.

Mathematical modeling uses real-world contexts; the model is an approximation, not the reality.

Types of models: linear, exponential, quadratic, trigonometric, and simulation models.

Assumptions: state simplifying assumptions explicitly; discuss their effect on results.

Validation: compare model predictions with data; assess reasonableness and limitations.

Representation: use graphs, tables, equations, and verbal descriptions to communicate the model.

Technology: spreadsheets, graphing calculators, and coding can support iteration and visualization.

Problem-solving strategies: draw a diagram, work backward, find a pattern, solve a simpler problem, guess and check.