12.2 Learning Progressions & the Concrete–Representational–Abstract Continuum

12.2 Learning Progressions & the CRA Continuum

Skill 6 of Competency 5 names the three modes explicitly — concrete, representational, and abstract — and asks you to analyze how understanding develops over time. This skill was added when Florida redeveloped test 025, so older prep material omits it entirely.

The CRA continuum

+---------------------------------------------------------------------------+
|   CONCRETE          ->     REPRESENTATIONAL    ->     ABSTRACT            |
|   "doing"                  "seeing"                   "symbolizing"       |
|                                                                           |
|   physical objects         drawings, diagrams,        numerals, variables,|
|   students manipulate      pictures of the objects    equations, formulas |
|                                                                           |
|   algebra tiles            a sketch of the tiles      x^2 + 5x + 6        |
|   fraction strips          a fraction bar diagram     3/4 + 1/8           |
|   two-color counters       a number line drawing      -3 - (-5) = 2       |
|   nets folded by hand      a drawing of the net       SA = 2B + Ph        |
+---------------------------------------------------------------------------+

Concrete — students physically handle objects. The manipulation carries the mathematical structure, and students can act on it before they can describe it.

Representational (semi-concrete) — students draw or interpret pictures of the same structure. This stage bridges to symbols because a drawing is already a representation rather than the thing itself, but it retains visible structure.

Abstract — students work with numerals, variables, and equations alone.

The order matters. Each stage builds meaning that the next stage compresses into notation. A student who has folded a net into a prism and then drawn one understands that surface area is the total of the faces; the formula SA = 2B + Ph becomes a shorthand for something already understood rather than a string to memorize.

[!IMPORTANT] The most common instructional error the exam tests is jumping to the abstract stage too early. The resulting student can execute notation but cannot detect an unreasonable result, cannot explain why a step is legal, and cannot reconstruct the procedure if a detail is forgotten. Every diagnostic pattern in section 11.2 traces back to this.

The continuum is not a one-way trip. Fluent learners move back to a drawing when a new problem type is unfamiliar, and returning to a concrete model is the standard remediation when abstract work breaks down.

Progressions within a topic

Fraction addition.

1. Concrete: combine fraction strips, discovering that unlike pieces cannot be combined directly.
2. Representational: draw partitioned bars and re-partition to a common size.
3. Abstract: find the least common denominator and add numerators.

The concrete stage is what makes the common denominator meaningful — you cannot combine thirds and fourths until both are expressed in twelfths, because you are counting pieces and the pieces must be the same size. A student who was given the LCD rule first has no answer to "why can't I just add the numerators and the denominators?"

Integer operations.

1. Concrete: two-color counters, forming and removing zero pairs.
2. Representational: number line jumps showing direction and magnitude.
3. Abstract: sign rules such as a − (−b) = a + b.

Solving equations.

1. Concrete: a balance scale with objects on both pans; removing the same amount from both sides keeps balance.
2. Representational: a drawn balance or bar model.
3. Abstract: the properties of equality applied symbolically.

The balance image is what makes "do the same thing to both sides" a necessity rather than a rule imposed by the teacher.

Area of a trapezoid.

1. Concrete: two congruent trapezoid cutouts rotated to form a parallelogram.
2. Representational: a diagram showing the composed parallelogram with base b₁ + b₂.
3. Abstract: A = ½(b₁ + b₂)h.

Progressions across grade levels

Progressions also run across years, and knowing the trajectory lets you locate where a gap originates.

• Additive to multiplicative reasoning. Early grades think additively ("it went up by 3"); middle grades must shift to multiplicative and proportional thinking ("it tripled"). Students stuck in additive reasoning answer "if 3 pencils cost $1.50, then 6 pencils cost $3.00" correctly by doubling but fail when the numbers are not multiples.
• Arithmetic to algebraic thinking. The equals sign shifts from "here comes the answer" to "these two expressions have the same value." A student who reads 8 + 4 = □ + 5 and writes 12 in the box holds the operational meaning and needs the relational one.
• Whole numbers to rational numbers. Properties that held for whole numbers fail: multiplication no longer always increases, division no longer always decreases, and there is no "next" number after 1/2.
• Specific to general. Numeric patterns become variable expressions, and particular figures become general theorems.

Using a progression diagnostically

Given a struggling student, locate the highest stage they hold securely and resume there.

A student solving 2x + 5 = 13 subtracts 5 from the left side only, obtaining 2x = 13, then divides to get x = 6.5. The student is applying operations to one side, which means the balance concept is not established. More symbolic practice will reinforce the error. The remediation is to return to the concrete stage with a balance model, so the student sees that removing weight from one pan alone tips the scale.

A student cannot simplify 3/6 + 1/6 but can correctly shade and combine sixths on a fraction bar. The representational stage is secure and the abstract stage is not, so instruction should explicitly connect the drawing to the symbols rather than restart at the concrete stage.

That second case is the finer judgment: the correct move is one stage back, not all the way back. Items offer both "return to concrete manipulatives" and "connect the drawing to the notation," and the second is right when the representational stage is already demonstrated.