11.2 Instructional Frameworks: CPA & Three-Part Lesson

11.2 Instructional Frameworks: CPA & Three-Part Lesson

Quick Summary: Effective mathematics instruction in Ontario relies on evidence-based instructional frameworks designed to build deep conceptual understanding before procedural fluency. The Concrete-Pictorial-Abstract (CPA) progression—grounded in Jerome Bruner's cognitive development theory—ensures students manipulate physical tools before moving to visual diagrams and symbolic algorithms. Teachers deliver this instruction using the Three-Part Math Lesson structure (Minds On, Action, Consolidation), incorporating differentiated instruction through open-ended questions ("Low Floor, High Ceiling") and parallel tasks.

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The Concrete-Pictorial-Abstract (CPA) Instructional Progression

The Concrete-Pictorial-Abstract (CPA) approach is a developmental framework that structures learning by introducing mathematical concepts through concrete hands-on experiences, transitioning to pictorial representations, and culminating in abstract symbolic manipulation.

graph LR
    C["1. CONCRETE STAGE<br/>(Enactive)<br/>Physical Manipulatives & Hands-on Tools"] --> P["2. PICTORIAL STAGE<br/>(Iconic)<br/>Visual Diagrams, Graphs & Models"] --> A["3. ABSTRACT STAGE<br/>(Symbolic)<br/>Numbers, Operations & Algebraic Symbols"]
    
    A -.->|"Revisit for Support / Verification"| P
    P -.->|"Grounding Conceptual Meaning"| C
    
    style C fill:#1e3a5f,color:#fff
    style P fill:#2d5a87,color:#fff
    style A fill:#c9a227,color:#1e3a5f

The Three Stages Breakdown

1. Concrete Stage (Enactive)

• Description: Students actively manipulate physical objects to model mathematical concepts directly. Physical handling provides tactile brain stimulation and grounds abstract ideas in real-world sensations.
• Manipulatives Used: Base-ten blocks (place value, multi-digit addition/subtraction), algebra tiles (polynomial operations, factoring), fraction strips/circles (equivalent fractions), relational rods (Cuisenaire rods for ratios), geoboards (perimeter and area), and pattern blocks (tessellations and spatial transformations).
• Example: To solve $2x + 3 = 11$, a student sets up 2 green tile bars ($x$) and 3 positive unit tiles on one side of a balance scale graphic, placing 11 positive unit tiles on the other side, physically removing 3 unit tiles from both sides.

2. Pictorial Stage (Iconic)

• Description: Students transition from physical objects to visual drawings, diagrams, charts, and spatial models that represent the physical objects.
• Visual Tools Used: Number lines (integer operations), bar models / tape diagrams (proportional reasoning and word problems), rectangular area grid models (multiplication of fractions and binomial expansion), array diagrams, and dot plots.
• Example: To model fraction multiplication $\frac{1}{2} \times \frac{3}{4}$, a student draws a rectangle, shades $\frac{3}{4}$ vertically with blue lines, shades $\frac{1}{2}$ horizontally with red lines, and counts the overlapping grid squares ($\frac{3}{8}$).

3. Abstract Stage (Symbolic)

• Description: Students represent mathematical ideas using abstract symbols, numerals, operational signs ($+, -, \times, \div$), and variables ($x, y, z$).
• Symbolic Tools Used: Mathematical equations, formulas ($A = lw, V = \pi r^2 h$), standard computational algorithms, and formal algebraic proofs.
• Example: Solving $2x + 3 = 11$ symbolically by writing $2x = 8 \implies x = 4$.

Critical Pedagogical Rule: The CPA model is not a linear one-way ladder to be permanently abandoned once abstract algorithms are introduced. Proficient mathematicians fluidly move back and forth between concrete, pictorial, and abstract representations when tackling complex, unfamiliar problems.

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The Three-Part Math Lesson Framework

In Ontario classrooms, problem-based mathematics lessons are structured using the Three-Part Math Lesson model. This framework shifts the teacher's role from lecturer to facilitator of student inquiry.

graph TD
    MINDS["PART 1: MINDS ON<br/>(10 - 15 Minutes)<br/>Activation, Hook & Goal Setting"]
    ACTION["PART 2: ACTION!<br/>(30 - 40 Minutes)<br/>Active Problem Solving & Exploration"]
    CONSOL["PART 3: CONSOLIDATION<br/>(10 - 15 Minutes)<br/>Bansho / Math Congress & Strategy Debrief"]
    
    MINDS --> ACTION --> CONSOL
    
    style MINDS fill:#1e3a5f,color:#fff
    style ACTION fill:#2d5a87,color:#fff
    style CONSOL fill:#c9a227,color:#1e3a5f

Detailed Breakdown of the Three Parts

Facilitating Effective Consolidation: Bansho and Math Congress

Consolidation is widely considered the most critical phase of the lesson. Ontario teachers utilize specific structured debrief techniques:

• Bansho (Board Writing): The teacher organizes student solution methods on the main whiteboard from left to right, ordered systematically by mathematical progression (e.g., from concrete/counting strategies $\rightarrow$ additive strategies $\rightarrow$ multiplicative/algebraic strategies). This visual arrangement allows students to see how their informal methods connect to formal mathematical algorithms.
• Math Congress: The teacher strategically selects 2 or 3 student solutions that highlight distinct mathematical approaches or common misconceptions, leading a focused class discussion around specific learning goals.

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Differentiated Instruction: Open-Ended Questions & Parallel Tasks

Ontario classrooms represent diverse readiness levels, language backgrounds, and learning profiles. To provide equitable access without tracking or lowering expectations, teachers utilize two primary differentiation strategies:

1. Open-Ended Questions ("Low Floor, High Ceiling")

Open-ended questions are tasks designed with multiple entry points (Low Floor) so that all students can begin solving them immediately using basic concepts, while offering extension opportunities (High Ceiling) that challenge advanced learners.

• Closed Task Example (Low Differentiation): "Calculate the perimeter of a rectangle with length 8 cm and width 4 cm." (Only one answer: $24\text{ cm}$; tests procedural recall).
• Open-Ended Task Example (High Differentiation): "The perimeter of a rectangle is 24 cm. What might its length, width, and area be? Find at least three different solutions and describe any patterns you notice."
  • Low Floor Entry: A student uses square tiles to build a $10 \times 2$ rectangle and counts sides to get $24\text{ cm}$.
  • High Ceiling Extension: A student explores fractional dimensions ($8.5 \times 3.5$), graphs length vs. area, and proves that a square ($6 \times 6$) maximizes the area ($36\text{ cm}^2$).

2. Parallel Tasks

Parallel tasks involve providing two distinct tasks (Option A and Option B) simultaneously. The tasks differ in numerical complexity or level of abstraction, but both address the exact same Big Idea and learning goal. All students choose their task, and the entire class participates in a unified consolidation debrief together.

Example of Parallel Tasks (Grade 7 - Proportional Reasoning)

• Learning Goal: Understand and apply unit rates to determine the best financial value.
• Task Option A (Integer Focus): "Store A sells 4 t-shirts for $40. Store B sells 6 t-shirts for $54. Which store offers the better value per shirt? Explain your reasoning."
• Task Option B (Decimal/Fraction Focus): "Store A sells 3.5 kg of apples for $8.75. Store B sells 2.2 kg of apples for $5.94. Which store offers the better value per kilogram? Explain your reasoning."
• Shared Consolidation Question: "Regardless of which option you solved, what mathematical steps did you take to compare the two rates? Why is finding the unit cost per single item helpful?"

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Classroom Scenario Application: Lesson Redesign

Traditional Math Lesson (Need Improvement): A Grade 6 teacher writes the formula $A = \frac{b \times h}{2}$ on the board, demonstrates two examples of calculating triangle area on the projector, and assigns 20 textbook exercises for individual seatwork.

Redesigned Ontario Three-Part CPA Lesson:

1. Minds On (12 min): The teacher displays a rectangle on grid paper with base 6 units and height 4 units ($Area = 24$). The teacher cuts the rectangle along its diagonal into two congruent right triangles. Students participate in a number talk estimating the area of one triangle.
2. Action (35 min - CPA & Parallel Tasks): Students work in pairs with grid paper and geoboards (Concrete/Pictorial).
   • Option A: Construct right and isosceles triangles on grid paper, enclose them in rectangles, and determine how triangle area relates to rectangle area.
   • Option B: Construct obtuse composite triangles on geoboards and derive the spatial relationship between base, height, and area.
   • Students observe that every triangle is exactly half of a corresponding parallelogram or rectangle.
3. Consolidation (13 min - Bansho): The teacher posts student grid paper solutions on the board from left to right:
   • Solution 1: Counting grid squares manually ($12$ squares).
   • Solution 2: Enclosing the triangle in a $6 \times 4$ rectangle, calculating $6 \times 4 = 24$, and dividing by 2.
   • Solution 3: Generalizing symbolically that $Area = \frac{b \times h}{2}$.
   • The teacher leads a discussion establishing why the factor of $\frac{1}{2}$ is universally required.