4.2 Evidence-Based Math & Writing Interventions

Evidence-Based Math & Writing Interventions

Academic intervention in school psychology extends well beyond reading. Deficits in mathematics and written expression represent profound barriers to school completion, postsecondary access, and vocational self-sufficiency. Just as in literacy, effective intervention in mathematics and writing demands explicit instructional architecture, cognitive strategy training, and high-density opportunities to respond (OTR). Practitioners must replace outdated, unempirical methods—such as rote drill without conceptual modeling or teaching misleading mathematical "key words"—with validated, cognitive-behavioral instructional frameworks (NASP Practice Model Domain 3).

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The Concrete-Representational-Abstract (CRA) Instructional Sequence

The Concrete-Representational-Abstract (CRA) sequence (also termed Concrete-Semiconcrete-Abstract [CSA]) is a tiered instructional methodology rooted in Jerome Bruner's cognitive developmental stages (enactive, iconic, and symbolic). Extensive research (Miller, Mercer, & Witzel) demonstrates that CRA significantly improves conceptual and procedural understanding for students with learning disabilities and math difficulties.

                    THE CRA INSTRUCTIONAL PROGRESSION

   [1. CONCRETE]            [2. REPRESENTATIONAL]            [3. ABSTRACT]
  Physical Manipulatives      2D Visual/Pictorial Models      Symbolic Math Notation
  • Base-ten blocks           • Tally marks, sketches         • Numerals: 24 x 3
  • Fraction tiles            • Strip diagrams, bar models    • Standard algorithms
  • Algebra tiles             • Number lines                  • Operational signs (+, -, x, /)
         │                              │                              │
         └──────────────────────────────┴──────────────────────────────┘
                         Scaffolded, Gradual Fading

The Three Stages of CRA

1. Concrete Stage (Enactive):
   • Method: The teacher models and students manipulate physical, 3-dimensional objects to solve mathematical problems.
   • Tools: Base-ten blocks (place value/regrouping), two-color counters (integer operations), fraction strips/circles (fraction equivalence), Cuisenaire rods, and geometric algebra tiles.
   • Purpose: Develops deep conceptual grounding before introducing formal written procedures.
2. Representational / Semi-Concrete Stage (Iconic):
   • Method: Physical objects are faded and replaced with 2-dimensional visual, pictorial, or graphic representations of the mathematical quantities.
   • Tools: Drawing dots, boxes, tally marks, bar models (Singapore math strip diagrams), array grids, and numbered number lines.
   • Purpose: Teaches students to create their own visual mental representations of mathematical relationships.
3. Abstract Stage (Symbolic):
   • Method: Visual drawings are faded, and students interact solely with symbolic numbers, operational symbols (+, −, ×, ÷), variables, and standard computational algorithms.
   • Transition Rule: Students should achieve 80% to 90% accuracy at the concrete and representational stages before advancing to purely abstract problems. Crucially, when an abstract error occurs, the student is prompted to drop back temporarily to representational sketching to self-correct.

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Schema-Based Instruction (SBI) for Word Problems

One of the most persistent, damaging pedagogical practices in elementary and middle school mathematics is teaching students to search for "key words" (e.g., teaching that "altogether" always means add, "left" always means subtract, or "more" always means add).

The Key Word Hazard: Consider this problem: "Elena has 35 baseball cards. She has 12 more cards than Marcus. How many cards does Marcus have?" A student using the keyword method sees "more" and incorrectly adds 35 + 12 = 47. In reality, Marcus has fewer cards (35 − 12 = 23). The keyword strategy fails because it bypasses mathematical comprehension in favor of superficial lexical matching.

The Schema-Based Instruction (SBI) Solution

Developed extensively by Asha Jitendra and Lynn Fuchs, Schema-Based Instruction (SBI) explicitly teaches students to identify the underlying semantic structure (schema) of mathematical word problems, map the problem elements onto visual schematic diagrams, and derive a corresponding mathematical equation.

                      PRIMARY SBI WORD PROBLEM SCHEMAS

  1. CHANGE SCHEMA (Dynamic)        2. GROUP SCHEMA (Static)       3. COMPARE SCHEMA (Relational)
     Beginning ± Change = End           Part 1 + Part 2 = Whole        Larger - Smaller = Difference
     ┌─────┐   ┌───┐   ┌─────┐         ┌─────┐   ┌─────┐   ┌─────┐      ┌──────┐   ┌──────┐   ┌──────┐
     │  B  │ ± │ C │ = │  E  │         │ P1  │ + │ P2  │ = │  W  │      │  L   │ - │  S   │ = │ Diff │
     └─────┘   └───┘   └─────┘         └─────┘   └─────┘   └─────┘      └──────┘   └──────┘   └──────┘

The Core Additive and Multiplicative Schemas

1. Change Problems (Dynamic):
   • An initial quantity is either increased or decreased over time by a dynamic event.
   • Mathematical Equation: Beginning Amount ± Change Amount = Ending Amount (B ± C = E).
2. Group / Combine Problems (Static):
   • Two or more distinct subsets or parts combine to form a static whole. There is no change over time.
   • Mathematical Equation: Part₁ + Part₂ = Whole (P₁ + P₂ = W).
3. Compare Problems (Relational):
   • A static comparison between two distinct sets to determine how much greater or lesser one quantity is than another.
   • Mathematical Equation: Larger Quantity − Smaller Quantity = Difference (L − S = D).
4. Multiplicative Schemas (Vary / Proportion & Multiplicative Comparison):
   • Vary Problem: A unit rate problem where a constant relationship exists between two quantities (e.g., 4 apples per basket × 6 baskets = 24 apples; Unit Rate × Quantity = Total).
   • Multiplicative Comparison: A quantity is a specified multiple of another (Comparison = Multiplier × Referent).

The FOPS Problem-Solving Heuristic

SBI trains students to navigate word problems using the four-step FOPS cognitive strategy:

• F — Find the problem type: Read the problem, ignore the numbers, identify the schema (Change, Group, Compare).
• O — Organize the information: Select the corresponding schema diagram and map the known quantities and unknown variable (?) into the visual structure.
• P — Plan to solve: Translate the completed visual diagram into a formal mathematical equation.
• S — Solve the problem: Execute the computation and verify if the numerical solution is mathematically reasonable.

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Math Fact Fluency: High-Density Retrieval Protocols

Computational automaticity is essential for higher-level mathematical problem solving. When single-digit addition, subtraction, or multiplication facts are not stored in long-term memory for automatic retrieval, working memory becomes severely congested.

1. Incremental Rehearsal (IR; Tucker, 1989; Burns et al.)

• Empirical Ratio: An evidence-based drill protocol maintaining a rigid ratio of 90% known facts to 10% unknown facts (9 knowns to 1 unknown).
• Folding Sequence Protocol:
  1. Assess a pool of flashcards to identify known facts (answered correctly within 2 seconds) and unknown facts.
  2. Select 9 known cards (K₁ through K₉) and 1 unknown card (U₁).
  3. Present in an expanding, folding sequence: $$U_1 \rightarrow K_1 \rightarrow U_1 \rightarrow K_1, K_2 \rightarrow U_1 \rightarrow K_1, K_2, K_3 \dots \rightarrow U_1 \rightarrow K_1 \dots K_9$$
  4. Once U₁ is mastered, it becomes K₁, K₉ is retired, and a new unknown (U₂) is introduced.
• Clinical Benefit: Yields near-zero error rates, builds high behavioral momentum, lowers math anxiety, and guarantees exceptional long-term consolidation.

2. Cover-Copy-Compare (CCC)

• Protocol: A self-managed, 5-step cognitive-behavioral fluency drill:
  1. Look: Student studies a solved math fact model (e.g., 7 × 8 = 56).
  2. Cover: Student covers the model with an index card.
  3. Copy: Student writes the problem and answer from memory on the blank side of the sheet.
  4. Uncover: Student removes the index card.
  5. Compare: Student evaluates their written answer against the original model.
  • Error Correction Drill: If correct, proceed to next item. If incorrect, the student immediately covers the model and writes the correct fact 3 consecutive times.
• Effectiveness: Highly efficient, requires minimal teacher time, and boasts strong empirical effect sizes across computation and spelling.

3. Explicit Timing & Taped Problems

• Explicit Timing: Students complete brief, timed computation sprints (1 to 2 minutes) with self-graphing of correct digits. Timing stimulates focus and overcomes computational hesitation.
• Taped Problems: Students listen to an audio recording that presents math facts with a brief, systematic latency pause before the audio announces the correct answer. The student races to write the answer before the recording provides it ("beat the tape"). The latency pause is systematically decreased (e.g., from 4 seconds down to 1 second) across sessions.

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Neurodevelopmental Presentation of Dyscalculia

Dyscalculia is a specific neurodevelopmental learning disability characterized by severe impairments in the acquisition of basic numerical and mathematical competencies.

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Evidence-Based Written Expression Interventions

Writing is the most cognitively complex academic task demanded of students, requiring the simultaneous coordination of motor transcription, orthographic retrieval, syntactic formulation, thematic organization, and metacognitive self-regulation.

               SELF-REGULATED STRATEGY DEVELOPMENT (SRSD)
   ┌─────────────────────────────────────────────────────────────┐
   │  Stage 1: Develop Background Knowledge (Activate pre-skills)│
   │  Stage 2: Discuss It (Examine baseline, set goals, commit)   │
   │  Stage 3: Model It (Teacher think-aloud & self-talk)         │
   │  Stage 4: Memorize It (Internalize mnemonics & steps)        │
   │  Stage 5: Support It (Scaffolded collaborative practice)     │
   │  Stage 6: Independent Performance (Autonomous writing)      │
   └─────────────────────────────────────────────────────────────┘

1. Self-Regulated Strategy Development (SRSD; Graham & Harris)

In Graham and Perin's (2007) Writing Next meta-analysis, SRSD studies produced an average effect size of 1.14, the largest of the strategy-instruction approaches. SRSD is therefore a leading evidence-based choice for written expression. It combines explicit cognitive strategy instruction with self-regulation techniques (goal-setting, self-instruction, self-monitoring, and self-reinforcement).

The 6 Stages of SRSD Instruction

1. Develop Background Knowledge: Review foundational concepts, assess student prerequisite skills (vocabulary, sentence structures), and introduce the target genre.
2. Discuss It: The teacher conferences with the student, examines baseline writing samples, discusses the benefits of writing strategies, establishes personal writing goals, and secures student buy-in.
3. Model It: The teacher writes a composition from scratch in front of the student, using explicit think-aloud modeling. Crucially, the teacher models internal self-regulatory statements:
   • Problem definition: "What is it I have to do here?"
   • Focusing attention: "Don't rush. Stick to my plan."
   • Coping: "I'm feeling stuck, but that's okay. I'll read what I have so far."
   • Self-reinforcement: "That was a great supporting reason!"
4. Memorize It: The student memorizes the mnemonic acronyms, steps, and self-talk prompts until they can recite them automatically.
5. Support It (Guided Practice): The teacher and student write compositions collaboratively. The teacher provides heavy scaffolding, prompt cards, and graphic organizers, gradually fading assistance as student mastery emerges.
6. Independent Performance: The student writes independently, utilizing the internalized strategy, self-monitoring their progress, and evaluating their finished composition against rubric benchmarks.

Core SRSD Strategy Mnemonics

• General Planning Strategy: POW
  • P — Pull apart the prompt / Pick my idea
  • O — Organize my notes (using graphic organizers)
  • W — Write and say more
• Persuasive / Opinion Writing Strategy: POW + TREE
  • T — Topic sentence (state your opinion clearly)
  • R — Reasons (provide at least 3 compelling reasons)
  • E — Explanations (elaborate and provide evidence for each reason)
  • E — Ending (wrap it up with a strong concluding thought)
• Advanced Argumentative Writing: STOP + DARE
  • S — Suspend judgment (brainstorm both sides)
  • T — Take a side
  • O — Organize ideas
  • P — Plan more
  • D — Develop topic sentence
  • A — Add supporting ideas
  • R — Reject the counterargument (refutation)
  • E — End with conclusion

2. Sentence Combining

• Empirical Status: Synthesized in Graham and Perin's (2007) Writing Next meta-analysis, sentence combining produced an average effect size of 0.50, while traditional decontextualized grammar instruction produced a negative effect (−0.32).
• Procedure: Students are explicitly taught to combine two or more simple, choppy "kernel sentences" into a sophisticated compound or complex sentence using connecting words, relative clauses, and participial phrases:
  • Kernels: The storm arrived. The storm was fierce. The storm knocked down trees.
  • Combined (Cued): The storm, which was fierce, arrived and knocked down trees. / Arriving fiercely, the storm knocked down trees.

3. Evidence-Based Spelling Interventions

• Cover-Copy-Compare (CCC) for Spelling: Identical 5-step self-monitoring protocol applied to orthographic spelling words.
• Phoneme-Grapheme Analysis & Morphemic Spelling Rules: Rather than assigning arbitrary weekly word lists, instruction explicitly teaches morphophonemic rules (e.g., the doubling rule when adding suffixes: hop → hopping vs. hope → hoping; Latin roots and Greek combining forms).

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Case Vignette: Applying Math & Writing Interventions

Student: Mateo, 5th grade.
Referral Problem: Severe failure in math word problems and written expression; teacher states Mateo "freezes during math tests and writes only one sentence for writing assignments."

Diagnostic Profile

• Math CBM (M-CAP): 4th percentile. Assessment reveals Mateo adds all numbers in word problems indiscriminately using a flawed keyword strategy. Single-digit fact retrieval is slow and inaccurate.
• Written Expression CBM: Total Words Written (TWW) = 14, Correct Writing Sequences (CWS) = 6 in 3 minutes. Writing consists of disjointed fragments without capitalization or punctuation.

Intervention Program

1. Mathematics Intervention (30 min, 3x/week):
   • Incremental Rehearsal (IR): Implemented for multiplication facts 0–9. Folding 1 unknown fact into 9 known facts. Fact retrieval rose from 12 correct digits to 38 correct digits per minute.
   • Schema-Based Instruction (SBI): Taught the Compare schema (Larger − Smaller = Difference) using the FOPS cognitive heuristic and schema diagrams. Eliminated the keyword guessing error.
2. Written Expression Intervention (30 min, 3x/week):
   • SRSD Instruction: Guided Mateo through the 6 stages of SRSD for persuasive writing using the POW+TREE mnemonic. The teacher explicitly modeled think-aloud self-talk during Stage 3.
   • Sentence Combining: 10 minutes of daily sentence combining exercises.

Clinical Outcome

After 12 weeks of intervention, Mateo's CWS increased from 6 to 28 sequences, and he independently composed a 5-paragraph persuasive essay utilizing POW+TREE. In math, his M-CAP word problem accuracy rose from 15% to 75% correct.