NBPTS AYA Math key terms

A repeating process of taking a real situation, simplifying it into a mathematical form, working within that form, interpreting results back in context, and revising the model. It emphasizes assumptions, validation, and iteration rather than a single right answer.

Deliberate choices about which features of a real situation to keep and which to ignore so a problem becomes tractable. Students should state assumptions and judge whether they are reasonable for the question being asked.

The effortful, sometimes uncomfortable thinking students do when a task is challenging but within reach. Teachers design for it and support it without removing the challenge, because struggle is where learning happens.

The level of thinking a task requires, ranging from memorization and routine procedures to procedures with connections and doing mathematics. Demand is set by the task and can be lowered or raised by how it is enacted.

Temporary supports, such as prompts, models, or partial worked examples, that help students succeed on a task they could not yet do alone. Supports are gradually removed as competence grows so students take ownership.

The range of tasks a learner can accomplish with guidance but not yet independently. Instruction aimed here, with appropriate support, produces growth more reliably than work that is too easy or too far beyond reach.

What a student already understands and can do that new learning depends on. Teachers diagnose it before instruction so they can build on accurate ideas and surface misconceptions that would otherwise interfere.

A stable but incorrect idea a student holds, often built from overgeneralizing a rule that works in limited cases. Misconceptions resist correction by simply being told the right answer and need tasks that create cognitive conflict.

Expressing the same mathematical idea through symbols, graphs, tables, diagrams, verbal descriptions, or physical models. Moving among representations builds flexible understanding and lets students connect procedures to meaning.

The ability to carry out procedures accurately, efficiently, and flexibly, and to choose an appropriate method. It is more than speed and should be developed alongside conceptual understanding rather than in isolation.

Grasping why a mathematical idea or procedure works, including the relationships among ideas. It supports transfer to new problems and helps students remember and adapt methods instead of relying on rote steps.

Applying knowledge learned in one context to a new or unfamiliar situation. It is the goal of durable learning and is supported by varied practice, explicit connections, and tasks that require choosing among strategies.

A clear statement of what students should know or be able to do by the end of a lesson or unit, tied to content and practice standards. It guides task selection, instruction, and what evidence will count as success.

Planning that starts from desired results, then determines acceptable evidence of learning, then designs learning experiences. It keeps instruction aligned to goals rather than to activities chosen first.

Student-facing descriptions of what a successful response looks like, often co-created and written in plain language. They make quality visible so students can self-assess and revise their work.

Choosing problems and activities that serve the learning goal and demand the intended level of thinking. Good selection balances accessibility, richness, and the opportunities for reasoning, discourse, and connection.

Managing time and sequence within a lesson so that opening, exploration, discussion, and closure each get adequate attention. Pacing protects time for sense-making rather than rushing to cover content.

Classroom talk in which students explain reasoning, compare strategies, justify claims, and question one another. It is structured by the teacher and moves understanding beyond answers toward argument and evidence.

Specific teacher prompts that extend student thinking, such as asking for reasoning, revoicing an idea, asking a student to restate another's thinking, or pressing for agreement. They distribute participation and deepen discussion.

Supporting a claim with valid reasoning, such as definitions, properties, examples, or a logical argument. It differs from giving an answer or citing authority and is central to proof and to classroom argument.

Gathering evidence of student understanding during learning and using it to adjust instruction or give feedback. Its purpose is to close the gap between current and desired performance, not to assign a grade.

Evaluation of student learning at the end of a unit or period, used to report achievement against standards. It should be aligned to the same goals and cognitive demand as the instruction it follows.

A short end-of-lesson task that yields quick evidence of whether students met the day's goal. Responses are reviewed to plan the next lesson and to identify students needing follow-up.

A pre-instruction check of relevant prior knowledge, skills, or misconceptions. Results inform planning, grouping, and the amount of review or extension a unit needs before new content begins.

Specific, timely information about what a student did well and what to do next, tied to success criteria. It is more useful than a score alone because it tells the learner how to improve.

A scoring guide that describes levels of performance on defined criteria. A well-built rubric makes expectations transparent, supports consistent judgment, and can be used by students for self- and peer-assessment.

Examining incorrect work to identify the reasoning behind the error rather than only marking it wrong. It turns mistakes into instructional information and helps students build durable, correct understanding.

Structuring classroom activity so all students, regardless of background or prior attainment, have access to rich mathematics and real opportunities to contribute. It includes attention to grouping, language, and who gets called on.

Using students' cultural backgrounds, experiences, and communities as resources for learning while holding high expectations for all. It connects mathematical tasks to students' lives and challenges deficit assumptions about learners.

The skills, knowledge, and practices families and communities already possess, such as measuring, budgeting, or planning. Teachers draw on these to anchor mathematics in meaningful contexts and to build relationships with families.

Proactively designing instruction with multiple means of representation, engagement, and action so diverse learners can access content. It reduces barriers by building flexibility into materials and tasks from the start.

Systematically examining one's own teaching, using evidence such as student work, lesson records, and feedback, to identify what worked and what to change. It is ongoing and focused on improving student learning.

A collaborative cycle in which teachers jointly plan a research lesson, observe it with a focus on student thinking, and revise it. The goal is shared learning about teaching, not evaluating the observed teacher.

A team of educators who meet regularly to examine evidence of student learning, share practice, and improve instruction together. Its work is collaborative, data-informed, and focused on results for students.

Building two-way partnerships with families so they can support learning at home and contribute information about their child. It is most effective when communication is regular, respectful, and free of jargon.

Adjusting content, process, product, or environment in response to student readiness, interests, and learning profiles. It aims to keep the same essential learning goal while varying the route to it.

Habits of mind such as making sense of problems, reasoning, constructing arguments, modeling, and using tools strategically. They describe how students engage with content and should be planned for, not left to chance.