Frameworks and Models Used in The Math

Mathematical thinking doesn't develop in a vacuum — it gets organized by frameworks. The models and structures explored here represent the major pedagogical and cognitive architectures that shape how mathematics is taught, sequenced, and assessed across K–12 and beyond. Understanding these frameworks matters because they explain why two classrooms covering the same topic can produce radically different outcomes.

Definition and scope

A framework, in the context of mathematics education, is a structured model that organizes how concepts are introduced, built upon, and connected. Frameworks operate at multiple levels: they govern curriculum sequencing, inform instructional choices, guide assessment design, and describe how students develop mathematical competence over time.

The scope here covers the dominant frameworks in use across US mathematics education, including cognitive models like the Concrete-Pictorial-Abstract (CPA) progression, curriculum frameworks like those embedded in the Common Core State Standards for Mathematics (CCSSM), and design models like Understanding by Design (UbD). It also includes assessment-oriented frameworks such as Bloom's Taxonomy (revised by Anderson and Krathwohl in 2001) and the Standards for Mathematical Practice outlined in CCSSM. Practitioners working across the full landscape of The Math encounter these frameworks constantly — often without knowing they're using one.

Core mechanics or structure

Concrete-Pictorial-Abstract (CPA)
Developed by psychologist Jerome Bruner and later formalized by Singapore's Ministry of Education into national curriculum design, the CPA framework moves learners through three stages: physical manipulation of objects, visual representation, and finally symbolic notation. A student counting blocks (concrete), drawing groups (pictorial), then writing "3 × 4 = 12" (abstract) is moving through all three phases. Singapore's 2020 Mathematics Curriculum Framework, published by the Ministry of Education Singapore, treats this sequence as foundational.

Standards for Mathematical Practice (SMPs)
The 8 Standards for Mathematical Practice, embedded in the Common Core State Standards (CCSSM, NGA/CCSSO, 2010), describe the behaviors of mathematically proficient students — things like constructing viable arguments, reasoning abstractly, and making use of structure. These aren't content standards; they're process standards. The distinction matters enormously in implementation.

Understanding by Design (UbD)
Grant Wiggins and Jay McTighe's backward design model, published by ASCD, organizes curriculum around 3 stages: identify desired results, determine acceptable evidence, then plan learning experiences. In mathematics, this flips the traditional lesson-plan sequence on its head — you start with what mastery looks like before choosing which problems to assign.

Mathematical Proficiency Strands
The National Research Council's Adding It Up (2001) identified 5 interlocking strands of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. These strands function as both an assessment lens and an instructional target. The National Academies Press hosts the full text freely.

Causal relationships or drivers

Frameworks don't emerge arbitrarily. The CPA sequence, for instance, gained institutional traction because of documented achievement gaps in procedural-only instruction — students who could execute algorithms but couldn't explain or transfer them. The cognitive science behind this is well-established: the Institute of Education Sciences Practice Guide Assisting Students Struggling with Mathematics (2021) identifies explicit instruction paired with visual representations as having "strong" evidence levels, the IES's highest rating category.

CCSSM's Standards for Mathematical Practice were driven in part by research showing that US students performed adequately on routine calculations but significantly below international peers on novel problem-solving — a pattern documented across multiple cycles of the Programme for International Student Assessment (PISA), administered by the OECD. The 2022 PISA results placed the United States at 34th among 81 participating education systems in mathematics (OECD PISA 2022 Results).

The UbD framework emerged as a response to what Wiggins and McTighe called "activity-focused" teaching — lessons organized around interesting tasks that lacked clear learning targets. The causal chain they identified: unclear goals → unfocused assessment → shallow retention.

Classification boundaries

Frameworks can be classified along two axes: scope (curriculum-level vs. lesson-level) and orientation (content vs. process).

These boundaries matter for implementation. Applying a lesson-level framework (like CPA) to a curriculum-scope problem produces mismatches. A school adopting CPA without a coherent scope-and-sequence still has a sequencing problem — the framework doesn't solve it. This is a persistent source of confusion in professional development work, explored further at The Math Professional Development for Teachers.

Tradeoffs and tensions

Fluency vs. Conceptual Depth
The most persistent tension in mathematics education frameworks is between procedural fluency and conceptual understanding. CCSSM explicitly names both as goals, and the 5 proficiency strands treat them as interlocking. But instructional time is finite. Spending more time on conceptual exploration early often means less time on fluency practice — and research on this tradeoff is genuinely mixed. The IES What Works Clearinghouse has rated conceptual-first approaches with strong evidence for some grade bands and insufficient evidence for others.

Coherence vs. Local Flexibility
National frameworks like CCSSM provide coherence across grade levels — a third-grade fraction standard is designed to connect to fifth-grade fraction operations. But 50 states implement standards differently, and many have modified or replaced CCSSM entirely. As of 2023, 41 states retained standards substantially aligned to CCSSM (Education Commission of the States, ECS State Education Policy), while others adopted independent frameworks. The coherence the model promises depends entirely on fidelity of implementation.

Assessment Alignment
UbD's backward design assumes that assessment drives instruction productively. In practice, when standardized tests don't align to the framework's assessment criteria, teachers face dual-target problems — preparing students for the framework's goals and for the test's goals simultaneously. This is documented extensively in NCTM's Principles to Actions (2014).

Common misconceptions

"CPA is only for young children."
The CPA sequence applies across grade levels. High school students working with polynomial functions benefit from manipulative modeling (algebra tiles, for instance) before symbolic abstraction. The misconception likely stems from the framework's prominence in elementary curriculum design, but Jerome Bruner's original framework was not grade-bounded.

"The 8 Standards for Mathematical Practice are optional add-ons."
The SMPs are not supplementary. CCSSM positions them as inseparable from content standards — the mathematical practices describe how students engage with content, not a separate enrichment layer. A classroom addressing content standards without the practices is, technically, only addressing half the standard.

"Backward design means teaching to the test."
UbD's backward design starts with desired understanding, not with test items. The distinction is that authentic evidence of understanding (performance tasks, transfer tasks) is different from a standardized test item. Conflating the two misreads the framework's intent entirely.

"Bloom's Taxonomy is a hierarchy of difficulty."
Bloom's revised taxonomy (Anderson & Krathwohl, 2001) describes levels of cognitive demand, not difficulty. A low-complexity task can require evaluation; a high-complexity problem can require only recall. The categories — remember, understand, apply, analyze, evaluate, create — describe cognitive processes, not content difficulty ratings.

Checklist or steps

Phases in applying the CPA sequence to a new concept:

  1. Identify the concrete manipulative that directly models the mathematical relationship (physical counters, fraction bars, base-ten blocks)
  2. Confirm the manipulative maps cleanly to the symbolic operation — not all objects model all operations accurately
  3. Introduce pictorial representations that preserve the structure of the manipulative (drawings, diagrams, number lines)
  4. Bridge from pictorial to symbolic by having students annotate their diagrams with notation
  5. Verify symbolic fluency does not precede pictorial understanding — sequence is not optional
  6. Revisit concrete and pictorial phases when a new complexity layer is introduced (e.g., moving from single-digit to multi-digit multiplication)

Phases in applying UbD to a mathematics unit:

  1. Identify the enduring understanding (what students should retain 5 years later)
  2. Write the essential questions that frame conceptual inquiry
  3. Define acceptable evidence — what performance demonstrates understanding
  4. Design formative checkpoints aligned to the evidence criteria
  5. Select and sequence learning experiences and problems
  6. Map instructional resources to specific evidence criteria, not to lesson topics

Reference table or matrix

Framework Scope Orientation Primary Source Grade Band
Concrete-Pictorial-Abstract (CPA) Lesson-level Process Bruner (1966); MOE Singapore (2020) PreK–12
Standards for Mathematical Practice Curriculum-level Process CCSSM, NGA/CCSSO (2010) K–12
Mathematical Proficiency Strands Curriculum-level Process NRC, Adding It Up (2001) K–8 focus
Understanding by Design (UbD) Lesson/Unit-level Content + Process Wiggins & McTighe, ASCD K–12
Bloom's Revised Taxonomy Lesson-level Process Anderson & Krathwohl (2001) K–16
NCTM Process Standards Curriculum-level Process NCTM (2000), Principles and Standards PreK–12
CCSSM Content Standards Curriculum-level Content NGA/CCSSO (2010) K–12

For a closer look at how these frameworks connect to specific instructional decisions, the core concepts reference and foundational principles pages address the underlying mathematical structures these models are built to serve.

References