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Progression Path Pedagogy

Overview

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.

Key Pedagogical Principles

1. Foundation Before Complexity

Students must master basic addition (sums ≤ 9) before learning carrying. This builds:

Number sense and fact fluency
Confidence with the addition operation
Mental calculation strategies

2. Graduated Difficulty

Three levels of regrouping difficulty:

0% regrouping (pAnyStart: 0) - All sums ≤ 9 (e.g., 3+4, 5+2)
50% regrouping (pAnyStart: 0.5) - Mixed practice
100% regrouping (pAnyStart: 1.0) - All problems require carrying

3. Scaffolding Cycle Pattern

For each new complexity level (digit count):

Full scaffolding - Ten-frames + carry boxes + place value colors
Fade scaffolding - Remove ten-frames, keep structure
Increase complexity - Add more digits, reintroduce scaffolding

4. Mastery-Based Progression

Students advance when they demonstrate:

Accuracy: 85-95% correct (varies by difficulty)
Volume: Minimum 15-20 problems attempted
Consistency: Sustained performance over multiple worksheets

Current Progression Path

Phase 0: Foundation (Steps 0-1)

Step 0: Basic Single-Digit Addition

Config: 1 digit, 0% regrouping, minimal scaffolding

pAnyStart: 0; // All sums ≤ 9
tenFrames: "never";
placeValueColors: "never";
carryBoxes: "never";

Sample Problems:

3 + 4 = 7
5 + 2 = 7
6 + 1 = 7
4 + 3 = 7

Mastery: 95% accuracy, 15 problems minimum

Rationale: Build foundational number sense and operation understanding without the cognitive load of regrouping.

Step 1: Mixed Single-Digit Practice

Config: 1 digit, 50% regrouping, conditional scaffolding

pAnyStart: 0.5; // Half need carrying
tenFrames: "never";
placeValueColors: "whenRegrouping";
carryBoxes: "whenRegrouping";

Sample Problems:

3 + 4 = 7 (no carrying)
8 + 7 = 15 (carrying) ← Carry box shown
5 + 2 = 7 (no carrying)
9 + 6 = 15 (carrying) ← Carry box shown

Mastery: 90% accuracy, 20 problems minimum

Rationale: Gradual introduction to carrying in mixed context. Students see both types of problems and begin to recognize when carrying is needed.

Phase 1: Single-Digit Carrying (Steps 2-3)

Step 2: Full Scaffolding (100% regrouping)

Config: 1 digit, 100% regrouping, full visual support

pAnyStart: 1.0; // All require carrying
tenFrames: "whenRegrouping"; // ← TEN-FRAMES INTRODUCED
placeValueColors: "always";
carryBoxes: "whenRegrouping";

Sample Problems: (all show ten-frames)

8 + 7 = 15 🔟🔟 (visual: 8 dots + 7 dots = full frame + 5 dots)
9 + 6 = 15 🔟🔟
7 + 8 = 15 🔟🔟

Mastery: 90% accuracy, 20 problems

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.

Step 3: Minimal Scaffolding

Config: 1 digit, 100% regrouping, ten-frames removed

pAnyStart: 1.0;
tenFrames: "never"; // ← SCAFFOLDING FADED
placeValueColors: "always";
carryBoxes: "whenRegrouping";

Mastery: 90% accuracy, 20 problems

Rationale: Students internalize the carrying procedure and no longer need visual aids. The carry boxes remain to support procedural memory.

Phase 2: Two-Digit Carrying (Steps 4-5)

Same scaffolding cycle, new digit range:

Step 4: 2 digits, full scaffolding (ten-frames RETURN for new complexity)
Step 5: 2 digits, minimal scaffolding (ten-frames fade)

Sample Problems (Step 4):

27 + 18 = 45 (ones: 7+8=15, carrying to tens)
35 + 29 = 64 (ones: 5+9=14, carrying to tens)

Rationale: When complexity increases (more digits), scaffolding returns temporarily. This supports learning the new format while applying known carrying skills.

Phase 3: Three-Digit Carrying (Steps 6-7)

Same pattern, 3 digits:

Step 6: 3 digits, full scaffolding
Step 7: 3 digits, minimal scaffolding

Design Rationale

Why Start with No Regrouping?

Research shows that:

Cognitive Load: Regrouping is a complex procedure. Students need to master basic addition first.
Number Sense: Understanding magnitude relationships (e.g., 7+3=10) supports later regrouping.
Confidence: Early success motivates continued practice.
Diagnostic: If students struggle with basic addition, regrouping will be impossible.

Why Mixed Practice (50%)?

The transition step (Step 1) serves multiple purposes:

Recognition Training: Students learn to identify when carrying is needed
Strategy Development: Seeing both types helps students develop conditional reasoning
Reduced Anxiety: Not every problem is hard, maintaining motivation
Real-World Realism: Actual practice mixes problem types

Why Ten-Frames?

Ten-frames are a research-validated manipulative that:

Visualize Regrouping: Clearly shows "making ten" (8 dots + 7 dots = full frame + 5)
Support Subitizing: Quick recognition of quantities up to 10
Bridge Abstract/Concrete: Connects symbolic notation to visual quantity
Align with Base-10: Naturally represents our number system

Example visualization:

8 + 7 = ?

[●●●●●]  ← Top frame (carry to next place)
[●●●●●]
[●●○○○]  ← Bottom frame (ones remaining)
[○○○○○]

8 dots + 7 dots = 10 dots (full frame) + 5 dots = 15

Why Fade Scaffolding?

Scaffolding fading is essential for:

Independence: Students must eventually work without aids
Efficiency: Visual aids slow down calculation
Transfer: Skills must work in different contexts (tests, real life)
Assessment: Teacher needs to verify internalized understanding

Future Extensions

Multi-Carry Path (Not Yet Implemented)

Steps 8-13 would teach carrying in multiple places:

157 + 268 (carries in ones AND tens)
789 + 456 (carries in ones AND tens AND hundreds)

Subtraction Path (Not Yet Implemented)

Similar progression for borrowing:

Basic subtraction (no borrowing)
Mixed practice
Full borrowing with hints
Fade borrowing hints

Testing and Validation

When implementing changes to the progression path:

Verify step numbering: Sequential, 0-based, no gaps
Check navigation: Each step's next/previous IDs are correct
Test mastery thresholds: Reasonable accuracy requirements (85-95%)
Validate configs: All displayRules are defined, operator is correct
User testing: Have real students attempt the progression

Implementation Notes

File: src/app/create/worksheets/addition/progressionPath.ts

Key Constants:

SINGLE_CARRY_PATH: Array of ProgressionStep objects
Each step has: id, stepNumber, technique, name, description, config, mastery criteria, navigation

Helper Functions:

getStepFromSliderValue(): Map UI slider (0-100) to step
getSliderValueFromStep(): Map step to slider position
findNearestStep(): Match config to closest step
getStepById(): Lookup step by ID

References

Ten-Frames: Van de Walle, J. A. (2004). Elementary and Middle School Mathematics
Scaffolding Fading: Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving
Mastery Learning: Bloom, B. S. (1968). Learning for Mastery
Cognitive Load: Sweller, J. (1988). Cognitive load during problem solving

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