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Practice Problem Generation System

Overview

The tutorial system includes an intelligent practice problem generator that creates math problems based on specific abacus calculation skills the user has mastered. This system ensures learners only encounter problems they can solve with their current skill level while providing targeted practice for newly learned techniques.

Core Concept: Skill-Based Problem Generation

Unlike traditional math problem generators, this system considers which specific abacus techniques a user has learned. The difficulty of a problem is determined by analyzing what calculation techniques are required to solve it, not just the magnitude of the numbers involved.

Key Principle

The skill required to solve a problem depends on what number you're adding to the current abacus state, not the sequence of numbers in the problem.

For example:

Adding 4 to a current value of 3 requires the "4 = 5 - 1" complement skill
Adding 4 to a current value of 1 only requires direct earth bead addition
The skill needed is determined by the addition step, not the final sum

Abacus Calculation Skills

Five Complements (Single Column Operations)

These skills are needed when adding a number would exceed the capacity of earth beads (4 beads max per column):

Skill	Description	When Required
4 = 5 - 1	Use heaven bead, remove 1 earth bead	Adding 4 when 1+ earth beads active
3 = 5 - 2	Use heaven bead, remove 2 earth beads	Adding 3 when 2+ earth beads active
2 = 5 - 3	Use heaven bead, remove 3 earth beads	Adding 2 when 3+ earth beads active
1 = 5 - 4	Use heaven bead, remove 4 earth beads	Adding 1 when 4 earth beads active

Ten Complements (Multi-Column Operations)

These skills are needed when adding would exceed 9 in the current column and require carrying to the next column:

Skill	Description	When Required
9 = 10 - 1	Carry to tens column, subtract 1 from current	Adding 9 causes column overflow
8 = 10 - 2	Carry to tens column, subtract 2 from current	Adding 8 causes column overflow
7 = 10 - 3	Carry to tens column, subtract 3 from current	Adding 7 causes column overflow
6 = 10 - 4	Carry to tens column, subtract 4 from current	Adding 6 causes column overflow
5 = 10 - 5	Carry to tens column, subtract 5 from current	Adding 5 causes column overflow
4 = 10 - 6	Carry to tens column, subtract 6 from current	Adding 4 causes column overflow
3 = 10 - 7	Carry to tens column, subtract 7 from current	Adding 3 causes column overflow
2 = 10 - 8	Carry to tens column, subtract 8 from current	Adding 2 causes column overflow
1 = 10 - 9	Carry to tens column, subtract 9 from current	Adding 1 causes column overflow

Basic Operations

Skill	Description
Direct Addition (1-4)	Add earth beads directly without complements
Heaven Bead (5)	Use the heaven bead to represent 5
Simple Combinations	Combine heaven and earth beads (6-9)

Practice Step Configuration

Basic Parameters

interface PracticeStep {
  id: string;
  title: string;
  description: string;

  // Problem generation settings
  problemCount: number; // How many problems to generate
  maxTerms: number; // Maximum numbers in one addition problem (2-5)

  // Skill-based constraints
  requiredSkills: SkillSet; // Skills user must know
  targetSkills: SkillSet; // Skills to specifically practice
  forbiddenSkills: SkillSet; // Skills user hasn't learned yet

  // Advanced constraints (optional)
  numberRange?: { min: number; max: number };
  sumConstraints?: { maxSum: number; minSum?: number };

  // Tutorial integration
  position?: number; // Where in tutorial flow this appears
}

Skill Set Definition

interface SkillSet {
  // Five complements (5-column operations)
  fiveComplements: {
    "4=5-1": boolean;
    "3=5-2": boolean;
    "2=5-3": boolean;
    "1=5-4": boolean;
  };

  // Ten complements (carrying operations)
  tenComplements: {
    "9=10-1": boolean;
    "8=10-2": boolean;
    "7=10-3": boolean;
    "6=10-4": boolean;
    "5=10-5": boolean;
    "4=10-6": boolean;
    "3=10-7": boolean;
    "2=10-8": boolean;
    "1=10-9": boolean;
  };

  // Basic operations
  basic: {
    directAddition: boolean; // Can add 1-4 directly
    heavenBead: boolean; // Can use heaven bead (5)
    simpleCombinations: boolean; // Can do 6-9 without complements
  };
}

Problem Generation Algorithm

1. Skill Analysis

For each potential problem, the generator:

Simulates solving the problem step-by-step on a virtual abacus
Identifies which skills would be required at each addition step
Compares required skills against the user's known skills
Only generates problems that use known skills or specifically target new skills

2. Problem Types

Practice Mode

Required Skills: All skills needed must be in user's known skill set
Purpose: Reinforce and practice mastered techniques
Example: If user knows "4=5-1" but not "3=5-2", only generate problems needing the first skill

Learning Mode

Target Skills: Generate problems that specifically require newly introduced skills
Purpose: Provide focused practice on a single new technique
Example: After teaching "3=5-2", generate problems that require this specific complement

Mixed Mode

Skill Range: Combine multiple known skills in single problems
Purpose: Integrate different techniques and build fluency
Example: Problems requiring both five complements and ten complements

3. Validation Process

Before presenting a problem to the user:

Skill Check: Verify all required skills are in allowed set
Difficulty Check: Ensure problem matches intended difficulty level
Diversity Check: Avoid repetitive patterns in generated problems
Sum Validation: Respect maximum sum constraints for target skill level

Tutorial Editor Integration

Practice Step Creation Workflow

Select Position: Choose where in tutorial flow the practice appears
Define Prerequisites: Select which skills user has learned by this point
Choose Target Skills: Pick specific skills to practice (optional)
Set Parameters: Configure problem count, max terms, number ranges
Preview Problems: Generate sample problems to verify configuration
Validate: Ensure all generated problems are solvable with allowed skills

Editor UI Components

Skill Selection Matrix

Visual grid showing:

✅ Known skills (green) - user has learned these
🎯 Target skills (blue) - specifically practice these
❌ Forbidden skills (red) - user hasn't learned yet
⚪ Unused skills (gray) - not relevant for this practice

Problem Preview Panel

Real-time display of:

Sample generated problems
Skill analysis for each problem
Warning indicators for invalid configurations

Validation Feedback

⚠️ Warnings for impossible skill combinations
❌ Errors for unsolvable problems
✅ Confirmation when configuration is valid

Implementation Examples

Example 1: Early Basic Practice

const basicPractice: PracticeStep = {
  id: "practice-basic-addition",
  title: "Practice: Basic Addition (1-4)",
  description: "Practice adding numbers 1-4 using only earth beads",
  problemCount: 12,
  maxTerms: 3,
  requiredSkills: {
    basic: {
      directAddition: true,
      heavenBead: false,
      simpleCombinations: false,
    },
    fiveComplements: {
      /* all false */
    },
    tenComplements: {
      /* all false */
    },
  },
  numberRange: { min: 1, max: 4 },
  sumConstraints: { maxSum: 9 },
};

Example 2: Five Complement Introduction

const complementPractice: PracticeStep = {
  id: "practice-four-equals-five-minus-one",
  title: "Practice: Using 4 = 5 - 1",
  description: "Practice the complement technique when adding 4",
  problemCount: 8,
  maxTerms: 3,
  requiredSkills: {
    basic: { directAddition: true, heavenBead: true, simpleCombinations: true },
    fiveComplements: { "4=5-1": true /* others false */ },
    tenComplements: {
      /* all false */
    },
  },
  targetSkills: {
    fiveComplements: { "4=5-1": true },
  },
  sumConstraints: { maxSum: 9 },
};

Example 3: Mixed Advanced Practice

const mixedPractice: PracticeStep = {
  id: "practice-mixed-complements",
  title: "Practice: Mixed Complement Operations",
  description: "Combine five complements and ten complements",
  problemCount: 15,
  maxTerms: 4,
  requiredSkills: {
    basic: {
      /* all true */
    },
    fiveComplements: {
      /* all true */
    },
    tenComplements: { "9=10-1": true, "8=10-2": true, "7=10-3": true },
  },
  sumConstraints: { maxSum: 99 },
};

Benefits for Learners

Progressive Difficulty: Never encounter problems requiring unknown skills
Targeted Practice: Focus on specific techniques needing reinforcement
Skill Integration: Gradually combine multiple techniques
Confidence Building: Success rate remains high through appropriate challenge level
Mastery Tracking: Clear progression through specific abacus skills

Benefits for Educators

Precise Control: Fine-grained control over what skills are practiced
Curriculum Alignment: Practice sessions match exact tutorial progression
Problem Validation: Automatic verification that problems are solvable
Adaptive Content: Problems automatically adjust to learner progress
Data Insights: Track which specific skills need more practice

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