feat(worksheets): add foundational steps to progression path
PEDAGOGICAL IMPROVEMENT: Students should master basic addition before learning regrouping/carrying. Added two prerequisite steps to the beginning of the progression path: New Steps: - Step 0: Basic single-digit addition (0% regrouping, pAnyStart: 0) - All sums ≤ 9 (3+4, 5+2, 6+1, etc.) - Minimal scaffolding (no ten-frames, no carry boxes, no colors) - 95% mastery threshold, 15 problems minimum - Step 1: Mixed single-digit practice (50% regrouping, pAnyStart: 0.5) - Half problems simple, half require carrying - Introduces carry boxes and conditional colors - 90% mastery threshold, 20 problems minimum Previous Progression: - Started at Step 0 with 100% regrouping (8+7, 9+6, all carrying) - Too advanced - kids need foundational number sense first New Progression: - Step 0-1: Foundation (no/mixed regrouping) - Step 2-3: Single-digit carrying (old steps 0-1, renumbered) - Step 4-5: Two-digit carrying (old steps 2-3, renumbered) - Step 6-7: Three-digit carrying (old steps 4-5, renumbered) Documentation: - Added .claude/PROGRESSION_PEDAGOGY.md with complete pedagogical rationale - Documents research basis for progression design - Explains ten-frames, scaffolding fading, mastery criteria - Includes implementation notes and future extensions Changes: - progressionPath.ts: Added 2 new steps at beginning - progressionPath.ts: Renumbered all subsequent steps (+2) - progressionPath.ts: Updated navigation IDs (previous/next) - .claude/PROGRESSION_PEDAGOGY.md: New comprehensive pedagogy doc Research basis: - Van de Walle (2004): Ten-frames for number sense - Wood, Bruner & Ross (1976): Scaffolding fading - Bloom (1968): Mastery learning - Sweller (1988): Cognitive load theory 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-Authored-By: Claude <noreply@anthropic.com>
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Thomas Hallock
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# Progression Path Pedagogy
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## Overview
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The mastery progression system guides students through addition skills using research-based pedagogical scaffolding. This document describes the improved progression path that starts with foundational skills before introducing regrouping/carrying.
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## Key Pedagogical Principles
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### 1. **Foundation Before Complexity**
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Students must master basic addition (sums ≤ 9) before learning carrying. This builds:
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- Number sense and fact fluency
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- Confidence with the addition operation
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- Mental calculation strategies
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### 2. **Graduated Difficulty**
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Three levels of regrouping difficulty:
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- **0% regrouping** (pAnyStart: 0) - All sums ≤ 9 (e.g., 3+4, 5+2)
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- **50% regrouping** (pAnyStart: 0.5) - Mixed practice
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- **100% regrouping** (pAnyStart: 1.0) - All problems require carrying
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### 3. **Scaffolding Cycle Pattern**
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For each new complexity level (digit count):
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1. **Full scaffolding** - Ten-frames + carry boxes + place value colors
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2. **Fade scaffolding** - Remove ten-frames, keep structure
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3. **Increase complexity** - Add more digits, reintroduce scaffolding
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### 4. **Mastery-Based Progression**
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Students advance when they demonstrate:
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- **Accuracy**: 85-95% correct (varies by difficulty)
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- **Volume**: Minimum 15-20 problems attempted
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- **Consistency**: Sustained performance over multiple worksheets
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## Current Progression Path
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### Phase 0: Foundation (Steps 0-1)
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#### Step 0: Basic Single-Digit Addition
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**Config**: 1 digit, 0% regrouping, minimal scaffolding
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```typescript
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pAnyStart: 0 // All sums ≤ 9
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tenFrames: 'never'
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placeValueColors: 'never'
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carryBoxes: 'never'
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```
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**Sample Problems**:
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- 3 + 4 = 7
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- 5 + 2 = 7
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- 6 + 1 = 7
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- 4 + 3 = 7
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**Mastery**: 95% accuracy, 15 problems minimum
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**Rationale**: Build foundational number sense and operation understanding without the cognitive load of regrouping.
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#### Step 1: Mixed Single-Digit Practice
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**Config**: 1 digit, 50% regrouping, conditional scaffolding
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```typescript
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pAnyStart: 0.5 // Half need carrying
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tenFrames: 'never'
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placeValueColors: 'whenRegrouping'
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carryBoxes: 'whenRegrouping'
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```
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**Sample Problems**:
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- 3 + 4 = 7 (no carrying)
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- **8 + 7 = 15** (carrying) ← Carry box shown
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- 5 + 2 = 7 (no carrying)
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- **9 + 6 = 15** (carrying) ← Carry box shown
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**Mastery**: 90% accuracy, 20 problems minimum
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**Rationale**: Gradual introduction to carrying in mixed context. Students see both types of problems and begin to recognize when carrying is needed.
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### Phase 1: Single-Digit Carrying (Steps 2-3)
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#### Step 2: Full Scaffolding (100% regrouping)
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**Config**: 1 digit, 100% regrouping, full visual support
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```typescript
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pAnyStart: 1.0 // All require carrying
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tenFrames: 'whenRegrouping' // ← TEN-FRAMES INTRODUCED
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placeValueColors: 'always'
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carryBoxes: 'whenRegrouping'
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```
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**Sample Problems**: (all show ten-frames)
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- **8 + 7 = 15** 🔟🔟 (visual: 8 dots + 7 dots = full frame + 5 dots)
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- **9 + 6 = 15** 🔟🔟
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- **7 + 8 = 15** 🔟🔟
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**Mastery**: 90% accuracy, 20 problems
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**Rationale**: Ten-frames provide concrete visual representation of "making ten" (e.g., 8+7: take 2 from 7 to make 10, then add 5 more = 15). This supports the conceptual understanding of regrouping.
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#### Step 3: Minimal Scaffolding
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**Config**: 1 digit, 100% regrouping, ten-frames removed
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```typescript
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pAnyStart: 1.0
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tenFrames: 'never' // ← SCAFFOLDING FADED
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placeValueColors: 'always'
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carryBoxes: 'whenRegrouping'
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```
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**Mastery**: 90% accuracy, 20 problems
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**Rationale**: Students internalize the carrying procedure and no longer need visual aids. The carry boxes remain to support procedural memory.
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### Phase 2: Two-Digit Carrying (Steps 4-5)
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**Same scaffolding cycle**, new digit range:
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- Step 4: 2 digits, full scaffolding (ten-frames RETURN for new complexity)
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- Step 5: 2 digits, minimal scaffolding (ten-frames fade)
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**Sample Problems (Step 4)**:
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- **27 + 18 = 45** (ones: 7+8=15, carrying to tens)
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- **35 + 29 = 64** (ones: 5+9=14, carrying to tens)
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**Rationale**: When complexity increases (more digits), scaffolding returns temporarily. This supports learning the new format while applying known carrying skills.
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### Phase 3: Three-Digit Carrying (Steps 6-7)
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**Same pattern**, 3 digits:
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- Step 6: 3 digits, full scaffolding
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- Step 7: 3 digits, minimal scaffolding
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## Design Rationale
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### Why Start with No Regrouping?
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Research shows that:
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1. **Cognitive Load**: Regrouping is a complex procedure. Students need to master basic addition first.
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2. **Number Sense**: Understanding magnitude relationships (e.g., 7+3=10) supports later regrouping.
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3. **Confidence**: Early success motivates continued practice.
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4. **Diagnostic**: If students struggle with basic addition, regrouping will be impossible.
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### Why Mixed Practice (50%)?
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The transition step (Step 1) serves multiple purposes:
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1. **Recognition Training**: Students learn to identify when carrying is needed
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2. **Strategy Development**: Seeing both types helps students develop conditional reasoning
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3. **Reduced Anxiety**: Not every problem is hard, maintaining motivation
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4. **Real-World Realism**: Actual practice mixes problem types
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### Why Ten-Frames?
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Ten-frames are a research-validated manipulative that:
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1. **Visualize Regrouping**: Clearly shows "making ten" (8 dots + 7 dots = full frame + 5)
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2. **Support Subitizing**: Quick recognition of quantities up to 10
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3. **Bridge Abstract/Concrete**: Connects symbolic notation to visual quantity
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4. **Align with Base-10**: Naturally represents our number system
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Example visualization:
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```
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8 + 7 = ?
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[●●●●●] ← Top frame (carry to next place)
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[●●●●●]
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[●●○○○] ← Bottom frame (ones remaining)
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[○○○○○]
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8 dots + 7 dots = 10 dots (full frame) + 5 dots = 15
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```
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### Why Fade Scaffolding?
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Scaffolding fading is essential for:
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1. **Independence**: Students must eventually work without aids
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2. **Efficiency**: Visual aids slow down calculation
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3. **Transfer**: Skills must work in different contexts (tests, real life)
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4. **Assessment**: Teacher needs to verify internalized understanding
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## Future Extensions
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### Multi-Carry Path (Not Yet Implemented)
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Steps 8-13 would teach carrying in multiple places:
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- 157 + 268 (carries in ones AND tens)
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- 789 + 456 (carries in ones AND tens AND hundreds)
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### Subtraction Path (Not Yet Implemented)
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Similar progression for borrowing:
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- Basic subtraction (no borrowing)
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- Mixed practice
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- Full borrowing with hints
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- Fade borrowing hints
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## Testing and Validation
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When implementing changes to the progression path:
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1. **Verify step numbering**: Sequential, 0-based, no gaps
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2. **Check navigation**: Each step's next/previous IDs are correct
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3. **Test mastery thresholds**: Reasonable accuracy requirements (85-95%)
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4. **Validate configs**: All displayRules are defined, operator is correct
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5. **User testing**: Have real students attempt the progression
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## Implementation Notes
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**File**: `src/app/create/worksheets/addition/progressionPath.ts`
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**Key Constants**:
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- `SINGLE_CARRY_PATH`: Array of ProgressionStep objects
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- Each step has: id, stepNumber, technique, name, description, config, mastery criteria, navigation
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**Helper Functions**:
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- `getStepFromSliderValue()`: Map UI slider (0-100) to step
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- `getSliderValueFromStep()`: Map step to slider position
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- `findNearestStep()`: Match config to closest step
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- `getStepById()`: Lookup step by ID
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## References
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- **Ten-Frames**: Van de Walle, J. A. (2004). Elementary and Middle School Mathematics
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- **Scaffolding Fading**: Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving
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- **Mastery Learning**: Bloom, B. S. (1968). Learning for Mastery
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- **Cognitive Load**: Sweller, J. (1988). Cognitive load during problem solving
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@ -43,19 +43,85 @@ export interface ProgressionStep {
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/**
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* Complete progression path for single-carry technique
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* This path demonstrates the scaffolding cycle pattern:
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* - Start with foundation (no regrouping)
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* - Introduce regrouping with support (mixed practice → full regrouping)
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* - Increase complexity (digit count) → reintroduce scaffolding (ten-frames)
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* - Fade scaffolding (remove ten-frames)
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* - Repeat
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*/
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export const SINGLE_CARRY_PATH: ProgressionStep[] = [
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// ========================================================================
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// PHASE 1: Single-digit carrying
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// PHASE 0: Foundation - Basic addition without regrouping
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// ========================================================================
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// Step 0: 1-digit with full scaffolding
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// Step 0: Basic single-digit addition (NO regrouping)
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{
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id: 'basic-addition-1d',
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stepNumber: 0,
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technique: 'basic-addition',
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name: 'Basic single-digit addition',
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description: 'Master simple addition without regrouping (3+4, 5+2, 6+1, etc.)',
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config: {
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digitRange: { min: 1, max: 1 },
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operator: 'addition',
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pAnyStart: 0, // 0% regrouping - all sums ≤ 9
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pAllStart: 0,
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displayRules: {
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carryBoxes: 'never', // Not needed yet
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answerBoxes: 'always',
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placeValueColors: 'never', // Keep it simple at first
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tenFrames: 'never', // Not needed yet
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problemNumbers: 'always',
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cellBorders: 'always',
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borrowNotation: 'never',
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borrowingHints: 'never',
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},
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interpolate: false,
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},
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masteryThreshold: 0.95, // Higher threshold for basics
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minimumAttempts: 15,
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nextStepId: 'mixed-addition-1d',
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previousStepId: null,
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},
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// Step 1: Mixed single-digit practice (SOME regrouping)
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{
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id: 'mixed-addition-1d',
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stepNumber: 1,
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technique: 'basic-addition',
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name: 'Mixed single-digit addition',
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description: 'Practice both simple and carrying problems together',
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config: {
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digitRange: { min: 1, max: 1 },
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operator: 'addition',
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pAnyStart: 0.5, // 50% regrouping - mix of easy and hard
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pAllStart: 0,
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displayRules: {
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carryBoxes: 'whenRegrouping', // Introduce carry boxes
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answerBoxes: 'always',
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placeValueColors: 'whenRegrouping', // Show colors when carrying
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tenFrames: 'never', // Not yet
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problemNumbers: 'always',
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cellBorders: 'always',
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borrowNotation: 'never',
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borrowingHints: 'never',
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},
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interpolate: false,
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},
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masteryThreshold: 0.9,
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minimumAttempts: 20,
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nextStepId: 'single-carry-1d-full',
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previousStepId: 'basic-addition-1d',
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},
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// ========================================================================
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// PHASE 1: Single-digit carrying with scaffolding
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// ========================================================================
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// Step 2: 1-digit with full scaffolding (100% regrouping)
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{
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id: 'single-carry-1d-full',
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stepNumber: 0,
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stepNumber: 2,
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technique: 'single-carry',
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name: 'Single-digit carrying (with support)',
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description: 'Learn carrying with single-digit problems and visual support',
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@ -79,13 +145,13 @@ export const SINGLE_CARRY_PATH: ProgressionStep[] = [
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masteryThreshold: 0.9,
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minimumAttempts: 20,
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nextStepId: 'single-carry-1d-minimal',
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previousStepId: null,
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previousStepId: 'mixed-addition-1d',
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},
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// Step 1: 1-digit with minimal scaffolding
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// Step 3: 1-digit with minimal scaffolding
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{
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id: 'single-carry-1d-minimal',
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stepNumber: 1,
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stepNumber: 3,
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technique: 'single-carry',
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name: 'Single-digit carrying (independent)',
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description: 'Practice carrying without visual aids',
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@ -116,10 +182,10 @@ export const SINGLE_CARRY_PATH: ProgressionStep[] = [
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// PHASE 2: Two-digit carrying (ones place only)
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// ========================================================================
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// Step 2: 2-digit with full scaffolding (SCAFFOLDING RETURNS!)
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// Step 4: 2-digit with full scaffolding (SCAFFOLDING RETURNS!)
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{
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id: 'single-carry-2d-full',
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stepNumber: 2,
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stepNumber: 4,
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technique: 'single-carry',
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name: 'Two-digit carrying (with support)',
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description: 'Apply carrying to two-digit problems with visual support',
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@ -146,10 +212,10 @@ export const SINGLE_CARRY_PATH: ProgressionStep[] = [
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previousStepId: 'single-carry-1d-minimal',
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},
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// Step 3: 2-digit with minimal scaffolding
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// Step 5: 2-digit with minimal scaffolding
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{
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id: 'single-carry-2d-minimal',
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stepNumber: 3,
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stepNumber: 5,
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technique: 'single-carry',
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name: 'Two-digit carrying (independent)',
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description: 'Practice two-digit carrying without visual aids',
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// PHASE 3: Three-digit carrying (ones place only)
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// ========================================================================
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// Step 4: 3-digit with full scaffolding (SCAFFOLDING RETURNS AGAIN!)
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// Step 6: 3-digit with full scaffolding (SCAFFOLDING RETURNS AGAIN!)
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{
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id: 'single-carry-3d-full',
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stepNumber: 4,
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stepNumber: 6,
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technique: 'single-carry',
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name: 'Three-digit carrying (with support)',
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description: 'Apply carrying to three-digit problems with visual support',
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@ -210,10 +276,10 @@ export const SINGLE_CARRY_PATH: ProgressionStep[] = [
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previousStepId: 'single-carry-2d-minimal',
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},
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// Step 5: 3-digit with minimal scaffolding
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// Step 7: 3-digit with minimal scaffolding
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{
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id: 'single-carry-3d-minimal',
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stepNumber: 5,
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stepNumber: 7,
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technique: 'single-carry',
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name: 'Three-digit carrying (independent)',
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description: 'Practice three-digit carrying without visual aids',
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