CRA Foundations Slides CRA Foundations
The Blueprint for Mathematical Accessibility
The "Glarb" Problem
Solve the following using the Glarbian system:
ΔΔ ⊕ ΔΞ = ?
How do you feel?
Confused?
Frustrated?
Incompetent?
"This is how a student with dyscalculia feels when we jump straight to abstract symbols without building a bridge."
The CRA Continuum
CONCRETE
Doing
Hands-on manipulation of physical objects. Building mental models through tactile feedback.
REPRESENTATIONAL
Seeing
Semi-concrete. Using pictures, tallies, or dots to stand in for physical objects.
ABSTRACT
Symbolic
Using numbers and mathematical symbols (+, −, ×) to represent concepts.
The "Why" for Special Education
Cognitive Load Reduction
Manipulatives offload working memory, allowing focus on conceptual logic rather than retrieval.
Multi-Sensory Encoding
Engages visual, tactile, and kinesthetic pathways simultaneously.
Error Pattern Analysis
Physical models make thinking visible, allowing teachers to pinpoint exact misconceptions.
Instructional Gaps
"Many students struggle not because they can't do the math, but because they are trapped in a cycle of abstract rules without concrete meaning."
Critiquing Practice
In your small groups, we will watch a video of 2nd-grade arithmetic instruction.
Observation Protocol:
1 Where along the CRA continuum is the teacher currently operating?
2 Does the student show signs of confusion (hesitation, guessing, incorrect application of rules)?
3 What concrete support could bridge this specific gap?
CRA Phase Worksheet CRA Phase Identification
Special Education: Instructional Blueprint Series
Name:
Date:
I. The Framework Matrix
Define each phase of the CRA continuum and identify one specific instructional tool used in that phase.
Phase Key Characteristics Example Tool Concrete Representational Abstract
II. Scenario Analysis
For each classroom scenario, identify which CRA phase is being utilized and suggest one way to scaffold it to the next phase.
Scenario A
A student is using base-ten blocks to model the number 25. They have 2 rods and 5 small cubes on their desk.
Current Phase:
Scaffold to Next Phase:
Scenario B
A teacher writes "25 + 14 =" on the whiteboard and asks students to find the sum without using any drawings or blocks.
Current Phase:
Scaffold Down (Concrete Support):
Scenario C
A student is drawing circles and tallies on their scratch paper to keep track of groups of ten while solving a multiplication problem.
Current Phase:
Link to Abstract:
III. Pedagogical Reflection
Why is it critical for students with Specific Learning Disabilities (SLD) in math to not skip the Representational phase?
Reflect on the "Glarb" simulation from class. How does that experience inform your future choice of manipulatives?
CRA Facilitation Guide CRA Facilitation Guide
Lesson 1: Foundations of the Framework
TEACHER RESOURCE
The Hook: Glarb Simulation
Instructions:
Display the "Glarb Problem" slide immediately. Do NOT explain the symbols. Give students 2 minutes to attempt the solution silently.
Key:
Δ = 1
Ξ = 10
⊕ = Addition
Problem: 2 + 11 = 13 (ΔΔΔΞ)
Debrief Questions:
"What was your immediate physical reaction to seeing the problem?"
"Why did your existing math knowledge fail to help you?"
"How would a handful of blocks (Concrete) have changed your experience?"
Instructional Critique
Prepare a 3-5 minute video of arithmetic instruction (suggested: multidigit addition without manipulatives). Direct students to look for the "Cognitive Gap."
The Mistake
Student follows a procedure (e.g., carrying a 1) but cannot explain why.
The Indicator
Hesitation, looking at the teacher for approval, "borrowing" from the wrong column.
The CRA Fix
Pause instruction. Introduce a concrete model to visualize the trade/regrouping.
Key Pedagogical Pillars
"The Representational stage is often the 'missing link.' It allows students to internalize the concrete movements before moving to the symbolic. Skipping it often leads to procedure-memory but no conceptual anchor."
"Precision in language is non-negotiable. Using 'trading' instead of 'carrying' or 'borrowing' maintains the connection to place value logic which is reinforced by the manipulatives."
Unit: CRA Mastery
Lesson 1: Foundations
Teacher Guide
Block Builders Slides Block Builders
Mastering Multi-Digit Operations with Base-Ten Blocks
The Language of Logic
AVOID
"Borrow" from the neighbor
"Carry" the one
"Move" the number
Why? These terms imply a temporary loan or a magic movement without place-value meaning.
USE
"Trade" 1 ten for 10 ones"Regroup" 10 ones as 1 ten"Compose" / "Decompose"
Why? These terms describe the physical equivalence shown by the blocks.
The Case of the Smaller-From-Larger Error
Student Work:
52
- 18
46
The Challenge:
Using your base-ten blocks, prove why the answer is NOT 46. Show exactly what happens to the "2" and the "8".
"The student likely did 8 minus 2 because they didn't understand the physical impossibility of taking 8 away from 2 without trading."
Explicit Scripting: Subtraction
Step 1: Set Up
"I have 5 tens and 2 ones. Can I take away 8 ones? No. I don't have enough."
Step 2: The Trade
"I will trade 1 ten rod for 10 ones. I still have 52, but now it looks like 4 tens and 12 ones."
Step 3: Calculate
"Now I take 8 ones away from 12 ones. I have 4 ones left. Then I take 1 ten away from 4 tens."
Note: Always model the physical movement of the trade alongside the verbal script .
Supporting Spatial Processing
Column Alignment
Use a Place Value Mat to provide rigid physical boundaries for rods and units.
Visual Anchors
The 10:1 ratio is visually obvious with blocks. For students with processing deficits, this visual proof is more reliable than memory.
Visual Aid: Place Value Mat Layout
Regrouping Scripts Handout Regrouping Scripts
Instructional Dialogue Protocols
Lesson 2 Tool
"Explicit instruction requires consistent, clear language that maps directly onto physical actions. Use these scripts during your peer-teaching practice to ensure students are building a place-value foundation, not just memorizing steps."
Explicit Script: Multi-Digit Addition
Step
Physical Action
Verbal Script
1. Set Up
Place blocks for both addends on mat.
"I have 3 tens and 7 ones (37). I want to add 2 tens and 5 ones (25). Let's put our ones together."
2. Check Ones
Count the total units in the ones column.
"How many ones do I have? 12. Can I keep 12 in the ones column? No, 10 or more means we must trade."
3. Regroup
Swap 10 units for 1 rod; move rod to tens.
"I am trading these 10 ones for 1 ten rod. I'll move this rod to the tens column. Now I have 2 ones left here."
4. Sum
Count total tens and ones.
"Now let's count: 6 tens and 2 ones. 37 plus 25 equals 62."
Explicit Script: Multi-Digit Subtraction
Step
Physical Action
Verbal Script
1. Minuend
Place blocks for the starting number only.
"I have 5 tens and 2 ones (52). I need to take away 1 ten and 8 ones (18). Let's start with the ones."
2. Analyze
Point to the units.
"Can I take away 8 ones when I only have 2? No. I need more ones. Where can I get them?"
3. Trade
Swap 1 rod for 10 units; place on units side.
"I will trade 1 ten rod for 10 ones. Now I have 4 tens and 12 ones. I still have 52, but now I have enough ones!"
4. Remove
Physically remove blocks from the mat.
"Now I take away 8 ones. I have 4 left. Now I take away 1 ten. I have 3 tens left. 52 minus 18 is 34."
Teacher Tip
Model the 'trade' by literally placing the rod in a container and taking 10 units out. Avoid just 'moving' things around; keep the exchange clear.
Spatial Cue
Keep 'taken away' blocks completely off the mat or in a separate 'trash' bowl to prevent visual clutter and confusion.
Base Ten Workshop Activity Base-Ten Workshop
Modeling Multi-Digit Operations
Workshop
02
Student:
Partner:
Instructions
Work with your partner to solve the following problems using base-ten blocks. One partner acts as the Teacher (using the Regrouping Scripts) while the other acts as the Learner . Switch roles for each problem. Document your 'trading' process below.
1
43 + 28
ADDITION
Trading Log:
We traded 10 ones for 1 ten.
No trade needed.
Describe the visual-spatial support you provided:
Explain how you aligned the blocks...
2
61 - 25
SUBTRACTION
Trading Log:
We traded 1 ten for 10 ones.
How many ones did you have after the trade? ________
Learner Confusion Point:
If the learner tried to do "5 - 1", what verbal prompt did you use?
3
127 + 85
ADDITION (HUNDREDS)
Trading Log:
Traded 10 ones for 1 ten.
Traded 10 tens for 1 hundred.
Modeling the 'Flat' (100):
Describe the physical trade of the ten rods for the hundred flat.
Peer Observation Feedback
Clarity of Language
Precision of Modeling
Error Correction
Array Architect Slides Array Architects
From Repeated Addition to Multiplicative Reasoning
The 12 x 13 Challenge
"How can you prove the product of 12 × 13 without using any numbers or the standard algorithm?"
The Problem with Counting:
Students with math disabilities often stay stuck in "Counting by Ones."
"1, 2, 3... 156."
The Geometric Solution:
Using the Area Model transforms a counting problem into a visualization problem.
Decomposing for Success
100
30
20
6
10 + 3 10 + 2
Partial Products
By breaking 12 × 13 into (10+2) × (10+3), students see four manageable rectangles.
Cognitive Access:
This prevents the "multiplication panic" that occurs when students lose track of large counts. They only need to know their 10s and basic facts.
Levels of Scaffolding
1. Discrete Objects
Using two-color counters in rows and columns. Hardest for processing (high visual noise).
2. Interlocking Units
Base-ten blocks or snap cubes. Units are visible but connected. Middle level of support.
3. Proportional Flats
The Area Model. Focuses on the size of the space, not individual counting. Closest to Abstract.
"When a student struggles, move them one level back. If they can't see the Area Model, let them feel the snap cubes."
Checking for Readiness
Before moving a student from Addition to Multiplication, ask:
Can they subitize small groups (see '3' without counting)?
Can they skip-count fluently (2, 4, 6, 8)?
Do they understand that one rod 'stands for' 10 ones?
Multiplication Modeling Worksheet Modeling Multiplication
Workshop 03: Area Models & Arrays
NAME:
Workshop Objective
Students will construct physical area models using base-ten blocks and tiles to decompose two-digit multiplication into partial products. Focus is on transitioning from counting-by-ones to multiplicative chunks.
1. Small Array (4 × 6)
LEVEL: DISCRETE
Build an array using 2-color counters.
Observation Task:
Observe your partner solving this. Do they count every counter (1, 2, 3...) or do they see the rows (6, 12, 18, 24)?
Partner's Strategy:
2. Decomposed Model (14 × 12)
LEVEL: PROPORTIONAL
Build the area model using 1 flat (100), rods (10s), and units (1s).
100
20
40
8
Diagram: (10+4) × (10+2)
Record Partial Products:
10 × 10 = ____
10 × 2 = ____
4 × 10 = ____
4 × 2 = ____
Total Product:
Scaffold Reflection
Imagine a student who can solve 12 + 12 + 12 but gets frustrated when they see 12 × 3. How does the physical area model help bridge that mental gap?
Multiplication Scaffolding Key Multiplication Scaffolding Key
Teacher Support: Lesson 3
Diagnostic: The 'Counting-by-Ones' Trap
"When students stay at the discrete level (counting every individual circle in an array), they aren't multiplying; they are doing high-volume addition. This leads to fatigue and high error rates."
Red Flags:
Touching every block/counter as they count.
Losing track of count after 20.
Inability to skip-count (e.g., cannot go from 6 to 12).
Viewing 12 × 3 as 12+12+12 but needing to draw all 36 dots.
The Scaffold:
Covering: Cover part of the array. Ask "If there are 6 in this row, and I have 3 rows total, how many are hidden?"Chunking: Use a rubber band to group rows.Slide to Area: Swap counters for base-ten rods to force the transition to tens.
Prompting the Area Model
When a student is stuck on 14 × 12:
Prompt 1:
"What is the 'nice' way to break 14? Can we use a ten?"
Prompt 2:
"What is the big area in the corner? 10 × 10 is...?"
Prompt 3:
"Where do these small 8 ones come from? Which two numbers met here?"
CRA Mastery Sequence
Lesson 3: Modeling Multiplication
Teacher Resource
Sketching Success Slides Sketching Success
Fading Physical Supports into Mental Models
Artistic vs. Schematic Drawings
Artistic (Avoid)
Drawing detailed base-ten blocks with all 10 unit lines, shadows, or 3D effects.
Too much cognitive load!
"If the drawing takes longer than the math, the scaffolding has become a barrier."
Schematic (Use)
Large Square = 100
Single Line = 10
Small Dot = 1
"Fast, clean, and symbolic enough to bridge to numerals."
The "X and Arrow" Technique
Subtraction Drawing: 32 - 15
Steps to Success:
Draw the total (minuend).
Cross out the rod to be traded.Draw an arrow to the ones column.
Draw 10 new ones.
Proceed with the calculation.
Criteria for Fading
Do not move to drawings until the student:
Can model the concept 3 days in a row with 100% accuracy.
Can explain the 'trade' verbally using precise language.
Has built 'muscle memory' and finds the blocks tedious.
Caution!
Premature fading leads to rule-following without meaning.
Teacher Challenge
"Pick up your drawing tool. We are going to practice the transition from physical blocks to 'dots and lines' across all operations."
Remember: Speed and Clarity > Artistic Accuracy.
Sketch Pad Practice Representational Sketch Pad
Bridging Concrete to Abstract
Date:
Initials:
Square = 100
Line = 10
Dot = 1
Task 1: Addition with Regrouping
45 + 37
Sketch Area:
Draw your schematic model here.
Process Description:
How did you show the trade of 10 ones for 1 ten in your drawing?
Task 2: Subtraction with Trading
62 - 28
Sketch Area:
Remember to use "X and Arrow" for trading.
Process Description:
Why is it important for a student with spatial processing issues to cross out the rod before drawing the new ones?
Task 3: Area Model Sketch
13 × 11
Sketch Area:
Show 100, 10s, and 1s separately.
Process Description:
Identify the four partial products shown in your drawing:
10 × 10 = ____
10 × 1 = ____
3 × 10 = ____
3 × 1 = ____
Fading Supports Checklist Guide Fading Support Checklist
Decision-Making Guide: Lesson 4
TEACHER GUIDE
"The goal of fading is to move the student from physical dependence to mental flexibility. Use this checklist to determine if a student is ready to transition from Concrete to Representational , or from Representational to Abstract ."
Transition 1: Concrete → Representational
Accuracy without Assistance
Student can build the model for three consecutive days with zero teacher prompts.
Verbal Fluency
Student can explain the 'trade' (e.g., "I trade 1 ten for 10 ones") while physically moving the blocks.
Physical Tedium
Student shows signs that the blocks are slowing them down or expresses a desire to 'just write it.'
Transition 2: Representational → Abstract
Schematic Consistency
Drawings are functional and efficient (dots/lines) rather than artistic or distracting.
Mental Model Retrieval
Student can describe the drawing they 'would' make before they actually pick up the pencil.
Dual-Coding Success
Student consistently records numerical notation alongside their drawings without being reminded.
When to backtrack:
If a student moves to drawings but starts making 'smaller-from-larger' errors or loses track of place value, re-introduce the physical blocks immediately. They have lost the conceptual anchor and are guessing at the procedure.
Algorithm Link Slides Algorithm Links
Connecting Visual Models to Abstract Notation
3
Simultaneous Instruction
The Error
Teaching the model on Monday, then teaching the algorithm on Friday. Students often fail to see they are the same thing.
"I do the blocks... then I do the real math."
The Fix
Dual Coding: Every physical move on the mat is immediately recorded on the paper as a numeral.
"The blocks explain the numbers."
The Hand-to-Pen Connection
Mat (Physical)
"I trade 1 ten for 10 ones..."
Algorithm (Abstract)
43 2
"...So I cross out the 4 and write 3."
Mini-Lesson Requirements
Explicit Verbal Script
Model demonstration
Side-by-side notation
Anticipated student error
Peer Feedback Focus
"Did the teacher pause to write the number at the exact moment the block was moved? If not, the link is broken."
Closing Thoughts
"CRA is not just a sequence of activities. It is the architectural support system for the student's internal logic."
When you teach the abstract algorithm, you are giving them the "why" that lives in their fingers and their eyes, and translates it to their pen.
Go forth and build accessible math.
Mini Lesson Planner Mini-Lesson Planner
CRA Sequence: Final Simulation
01
Target Skill & Error Analysis
Mathematical Operation:
e.g., Subtraction with regrouping...
Common Student Misconception:
What error pattern will you address?
02
Explicit Instruction Script
Write exactly what you will say. Use precise vocabulary (trade, regroup, decompose) and plan your simultaneous moves (Mat & Paper).
I DO
WE DO
03
Scaffolding & Fading
Concrete Tool Selection:
Representational Drawing:
Sketch your schematic model here...
Abstract Notation Check:
How will you physically point to the numeral and the block at the same time?
Final Checklist
Trading bowl/mat is ready
Script avoids 'borrow/carry'
Algorithm is visible to 'learner'
Anticipated error is modeled
CRA Mastery Rubric CRA Mastery Rubric
Final Simulation Assessment
MAX POINTS
20
Candidate:
Score:
Criterion Exemplary (4) Developing (2) Needs Revision (0-1) Precise Mathematical Language Uses 'trade', 'regroup', and 'decompose' consistently. Zero use of 'borrow' or 'carry'. Uses precise terms but slips occasionally into procedural jargon. Relies on 'borrow/carry' or other non-place-value language. Dual Coding & Timing Records numerals on paper at the exact moment the block is moved. Flawless synchronization. Finishes the block model first, then writes the whole algorithm. Connection is delayed. Fails to connect the model and algorithm side-by-side. C/R Fidelity Modeling/Drawing is physically accurate and supports the logic of the operation. Minor errors in modeling or disorganized placement of blocks/drawings. Modeling is incorrect or contributes to learner confusion. Error Correction Identifies student error immediately and redirects using the physical model as proof. Identifies error but corrects verbally without referring back to the model. Misses the student error or provides incorrect feedback. Explicit Instruction Flow I Do-We Do-You Do cycle is clear. Scaffolding is evident and systematic. Instruction is a bit rushed; jumps too quickly from teacher-model to independent work. Instructional sequence is disorganized or missing key components.
Instructor Feedback & Next Steps
Strength of Practice:
Critical Revision Area:
Faculty Evaluator Signature
