高等数学错题深度分析Skill

针对高等数学错题进行错误模式识别、知识点关联分析和个性化学习路径推荐，助力学习效果提升。

Security Scan

This instruction-only skill is internally consistent with its stated purpose (analyzing calculus errors and generating personalized study plans) and requests no unusual installs or credentials, but you should pay attention to student data privacy when using it.

Purpose & Capability

The name/description (高等数学错题深度分析) matches the SKILL.md content: analysis, error-book generation, and intervention recommendations. There are no unrelated required binaries, env vars, or config paths.

Instruction Scope

SKILL.md defines functions and the expected inputs/outputs (e.g., student_id, error_data) and contains example content and example resource URLs. It does not instruct the agent to access system files, credentials, or external endpoints beyond example resource links. However, the instructions do not specify how student data is supplied, stored, or transmitted—this is an operational/privacy consideration rather than an incoherence.

Install Mechanism

No install spec and no code files—this is instruction-only, so nothing is downloaded or written to disk by the skill itself.

Credentials

The skill declares no environment variables, credentials, or config paths. The parameters it expects (student_id, error_data, etc.) are appropriate for its stated purpose.

Persistence & Privilege

always is false and the skill does not request elevated persistence or modify other skills/settings. Autonomous invocation is allowed by default but not combined with other red flags.

Assessment

This skill appears coherent and low-risk from a supply-chain perspective because it is instruction-only and asks for no credentials. Before installing or using it, consider: (1) privacy—the skill processes student identifiers and error data: ensure you supply anonymized data or have consent and a secure data-handling process; (2) storage and transmission—decide where analysis results will be stored and who can access them; (3) external resources—example URLs in returned data are placeholders; ensure the agent does not automatically fetch or transmit student data to third-party sites without explicit approval; (4) validation—test outputs on known examples to confirm the recommendations are pedagogically appropriate; and (5) logging/retention—confirm how long sensitive data is retained in your system. If you need the skill to operate on real student records, require a documented data flow and appropriate access controls before enabling it.

calculus-error-analyzer - 高等数学错题深度分析Skill

概述

专门针对高等数学错题的深度分析系统，通过错误模式识别、知识点关联分析、学习路径优化，实现从错题到学习改进的智能转化。

核心功能

1. 错误模式深度挖掘

• 错误类型分类: 概念错误、计算错误、逻辑错误、理解偏差
• 错误根源分析: 追溯错误的知识点根源和思维误区
• 错误模式聚类: 发现学生个性化的错误模式
• 错误演变追踪: 分析错误随时间的演变规律

2. 个性化错题本生成

• 智能错题归类: 按知识点、错误类型、难度自动分类
• 错题标签系统: 多维标签标记错题特征
• 错题难度评估: 评估错题的典型性和重要性
• 错题复习计划: 基于遗忘曲线的智能复习安排

3. 学习路径优化

• 薄弱点诊断: 精准定位知识体系中的薄弱环节
• 学习路径推荐: 基于错误分析的个性化学习路径
• 干预策略生成: 针对不同错误类型的教学干预
• 进步预测: 基于错误纠正的进步可能性预测

工具定义

analyze_error_patterns

深度分析错误模式

参数：

• error_data (array): 错误数据列表
• analysis_level (string): 分析级别，"basic"、"standard"、"deep"
• include_causes (boolean): 是否包含错误原因分析
• generate_solutions (boolean): 是否生成解决方案

返回：

{
  "analysis_id": "ea_001",
  "summary": {
    "total_errors": 24,
    "unique_error_types": 8,
    "most_common_error": "积分公式记错",
    "error_frequency": 6
  },
  "error_categories": [
    {
      "category": "概念理解错误",
      "count": 10,
      "percentage": 41.7,
      "subtypes": [
        {
          "subtype": "定理条件混淆",
          "examples": ["误用罗尔定理条件", "拉格朗日定理应用错误"],
          "root_cause": "对定理成立条件理解不深",
          "recommendation": "重点复习定理的几何意义和适用条件"
        }
      ]
    },
    {
      "category": "计算执行错误",
      "count": 8,
      "percentage": 33.3,
      "subtypes": [
        {
          "subtype": "符号运算错误",
          "examples": ["正负号错误", "括号展开错误"],
          "root_cause": "计算粗心，检查习惯不好",
          "recommendation": "建立计算检查清单，放慢计算速度"
        }
      ]
    }
  ],
  "knowledge_gaps": [
    {
      "topic": "微分中值定理",
      "gap_level": "严重",
      "affected_questions": 5,
      "prerequisite_topics": ["函数连续性", "导数概念"],
      "remediation_path": [
        "第一步：复习函数连续性定义",
        "第二步：理解导数几何意义",
        "第三步：学习中值定理证明",
        "第四步：练习典型应用题"
      ]
    }
  ],
  "personalized_insights": {
    "learning_style_issues": "倾向于记忆公式而非理解推导",
    "thinking_patterns": "在证明题中容易跳过关键步骤",
    "time_management": "复杂题目后期容易出现计算错误",
    "confidence_level": "中等偏下，需要成功体验提升"
  }
}

generate_personalized_error_book

生成个性化错题本

参数：

• student_id (string): 学生ID
• time_range (string): 时间范围，"week"、"month"、"all"
• organization_method (string): 组织方式，"by_topic"、"by_error_type"、"by_difficulty"
• include_explanations (boolean): 是否包含详细解析

返回：

{
  "error_book_id": "eb_001",
  "student_info": {
    "student_id": "stu001",
    "name": "张三",
    "total_errors_collected": 42,
    "time_period": "2026-03-01 至 2026-04-15"
  },
  "error_book_structure": {
    "sections": [
      {
        "section_title": "第一章 函数与极限",
        "error_count": 8,
        "priority": "高",
        "topics": [
          {
            "topic": "函数极限计算",
            "error_examples": [
              {
                "original_question": "求lim(x→0) (sinx/x)",
                "student_answer": "0",
                "correct_answer": "1",
                "error_type": "重要极限记错",
                "detailed_explanation": "这是第一个重要极限，值为1...",
                "similar_questions": [
                  "lim(x→0) (tanx/x)",
                  "lim(x→0) (arcsinx/x)"
                ],
                "learning_resources": [
                  {
                    "type": "video",
                    "title": "重要极限的几何解释",
                    "url": "https://example.com/video1"
                  }
                ]
              }
            ],
            "mastery_exercises": [
              {
                "exercise": "计算lim(x→0) (1-cosx)/x²",
                "hint": "使用半角公式或洛必达法则",
                "answer": "1/2"
              }
            ]
          }
        ]
      }
    ]
  },
  "review_schedule": {
    "spaced_repetition": [
      {
        "review_date": "2026-04-16",
        "topics": ["函数极限", "导数定义"],
        "estimated_time": 30
      },
      {
        "review_date": "2026-04-20",
        "topics": ["微分中值定理"],
        "estimated_time": 45
      }
    ],
    "review_strategies": {
      "immediate": "24小时内复习新错题",
      "short_term": "3天后第二次复习",
      "long_term": "每周系统性回顾"
    }
  },
  "progress_tracking": {
    "error_reduction_rate": 35.2,
    "mastery_improvement": [
      {"topic": "导数计算", "from": 65, "to": 82},
      {"topic": "积分应用", "from": 58, "to": 71}
    ],
    "confidence_change": "显著提升"
  }
}

recommend_intervention_strategies

推荐教学干预策略

参数：

• error_analysis (object): 错误分析结果
• student_profile (object): 学生画像
• intervention_type (string): 干预类型，"immediate"、"short_term"、"long_term"
• available_resources (array): 可用教学资源

返回：

{
  "intervention_plan_id": "ip_001",
  "student_profile_summary": {
    "learning_style": "视觉型学习者",
    "motivation_level": "中等",
    "self_efficacy": "需要提升",
    "time_availability": "每周5-7小时"
  },
  "immediate_interventions": [
    {
      "strategy": "概念澄清会议",
      "description": "一对一讲解微分中值定理的几何意义",
      "duration": 30,
      "materials": [
        "GeoGebra动态演示文件",
        "定理证明动画视频"
      ],
      "expected_outcome": "理解定理条件和几何解释"
    },
    {
      "strategy": "计算规范化训练",
      "description": "建立标准计算步骤检查表",
      "duration": 20,
      "materials": ["计算检查清单", "典型错误案例集"],
      "expected_outcome": "减少符号运算错误50%"
    }
  ],
  "short_term_strategies": [
    {
      "strategy": "错题重做计划",
      "description": "系统性重做最近10个错题",
      "schedule": "每天2题，连续5天",
      "monitoring": "每天提交重做结果",
      "success_criteria": "正确率达到90%"
    },
    {
      "strategy": "同伴学习小组",
      "description": "与同水平同学组成学习小组",
      "activities": ["互相批改作业", "讨论解题思路", "分享学习资源"],
      "frequency": "每周2次，每次60分钟"
    }
  ],
  "long_term_development": [
    {
      "strategy": "数学思维训练",
      "description": "培养严谨的数学推理习惯",
      "activities": [
        "每周完成1道证明题",
        "学习数学证明写作规范",
        "阅读数学史相关材料"
      ],
      "timeline": "8周计划",
      "milestones": [
        "第2周：掌握基本证明结构",
        "第4周：能独立完成中等难度证明",
        "第8周：形成系统的数学思维"
      ]
    }
  ],
  "resource_recommendations": {
    "targeted_practice": [
      {
        "resource_type": "练习册",
        "name": "微分中值定理专项练习",
        "difficulty": "基础到提高",
        "estimated_time": "4小时"
      }
    ],
    "conceptual_understanding": [
      {
        "resource_type": "视频课程",
        "name": "微积分核心概念可视化",
        "duration": "3小时",
        "focus": "几何直观理解"
      }
    ],
    "motivational": [
      {
        "resource_type": "阅读材料",
        "name": "数学之美：从微积分看世界",
        "purpose": "激发学习兴趣，理解数学应用"
      }
    ]
  },
  "monitoring_and_evaluation": {
    "key_metrics": [
      "错题重做正确率",
      "同类错误重复率",
      "学习时间投入",
      "自我效能感变化"
    ],
    "evaluation_schedule": [
      {"time": "1周后", "focus": "计算错误减少情况"},
      {"time": "2周后", "focus": "概念理解提升"},
      {"time": "1月后", "focus": "综合能力进步"}
    ],
    "adjustment_criteria": "如果2周后进步小于20%，调整干预策略"
  }
}

使用示例

示例1：深度错误模式分析

# 分析学生错误模式
openclaw skill calculus-error-analyzer analyze_error_patterns \
  --error-data '[{"question":"求导数","error":"链式法则应用错误"},{"question":"计算积分","error":"积分公式记错"}]' \
  --analysis-level "deep" \
  --include-causes true \
  --generate-solutions true

示例2：生成个性化错题本

# 为学生生成月度错题本
openclaw skill calculus-error-analyzer generate_personalized_error_book \
  --student-id "stu001" \
  --time-range "month" \
  --organization-method "by_topic" \
  --include-explanations true

示例3：推荐干预策略

# 基于错误分析推荐教学干预
openclaw skill calculus-error-analyzer recommend_intervention_strategies \
  --error-analysis '{"error_categories":[{"category":"概念错误","count":10}]}' \
  --student-profile '{"learning_style":"visual","motivation":"medium"}' \
  --intervention-type "comprehensive" \
  --available-resources '["geogebra","video_course","practice_sheets"]'

错误分析模型

错误分类体系

class ErrorTaxonomy:
    """高等数学错误分类体系"""
    
    def __init__(self):
        self.categories = {
            'conceptual_errors': {
                'definition_misunderstanding': {
                    'description': '概念定义理解错误',
                    'examples': ['极限定义混淆', '导数概念误解'],
                    'severity': '高'
                },
                'theorem_misapplication': {
                    'description': '定理条件或应用错误',
                    'examples': ['误用中值定理', '洛必达法则滥用'],
                    'severity': '高'
                },
                'logical_fallacies': {
                    'description': '逻辑推理错误',
                    'examples': ['循环论证', '充分必要条件混淆'],
                    'severity': '中'
                }
            },
            'computational_errors': {
                'algebraic_mistakes': {
                    'description': '代数运算错误',
                    'examples': ['符号错误', '因式分解错误'],
                    'severity': '中'
                },
                'formula_misremembering': {
                    'description': '公式记忆错误',
                    'examples': ['积分公式记错', '导数公式错误'],
                    'severity': '中'
                },
                'procedural_errors': {
                    'description': '计算步骤错误',
                    'examples': ['积分上下限错误', '变量替换错误'],
                    'severity': '低'
                }
            },
            'strategic_errors': {
                'method_selection': {
                    'description': '解题方法选择不当',
                    'examples': ['复杂方法代替简单方法', '方法不适用'],
                    'severity': '中'
                },
                'problem_analysis': {
                    'description': '问题分析不充分',
                    'examples': ['未识别关键条件', '误解问题要求'],
                    'severity': '高'
                }
            }
        }
    
    def classify_error(self, error_description, context):
        """分类错误类型"""
        for category, subcategories in self.categories.items():
            for subtype, config in subcategories.items():
                if self.matches_pattern(error_description, config['examples']):
                    return {
                        'category': category,
                        'subtype': subtype,
                        'description': config['description'],
                        'severity': config['severity'],
                        'confidence': self.calculate_confidence(error_description, config)
                    }
        return {'category': 'unknown', 'subtype': 'other'}

错误根源分析

class RootCauseAnalyzer:
    """错误根源分析引擎"""
    
    def analyze_root_cause(self, error_instance, student_history):
        """分析错误根本原因"""
        causes = []
        
        # 1. 知识点缺陷分析
        knowledge_gaps = self.identify_knowledge_gaps(error_instance)
        if knowledge_gaps:
            causes.append({
                'type': 'knowledge_gap',
                'description': '相关知识点掌握不足',
                'specific_gaps': knowledge_gaps,
                'evidence': self.find_evidence(student_history, knowledge_gaps)
            })
        
        # 2. 思维习惯分析
        thinking_patterns = self.analyze_thinking_patterns(error_instance, student_history)
        if thinking_patterns:
            causes.append({
                'type': 'thinking_habit',
                'description': '不良思维习惯',
                'patterns': thinking_patterns,
                'frequency': self.calculate_frequency(student_history, thinking_patterns)
            })
        
        # 3. 心理因素分析
        psychological_factors = self.assess_psychological_factors(student_history)
        if psychological_factors:
            causes.append({
                'type': 'psychological',
                'description': '心理因素影响',
                'factors': psychological_factors,
                'impact_level': self.estimate_impact(psychological_factors)
            })
        
        # 4. 学习策略分析
        strategy_issues = self.evaluate_learning_strategies(student_history)
        if strategy_issues:
            causes.append({
                'type': 'strategy',
                'description': '学习策略不当',
                'issues': strategy_issues,
                'recommendations': self.suggest_strategy_improvements(strategy_issues)
            })
        
        return {
            'primary_cause': self.identify_primary_cause(causes),
            'contributing_factors': causes,
            'interdependencies': self.analyze_interdependencies(causes)
        }

错题本智能生成

class ErrorBookGenerator:
    """个性化错题本生成器"""
    
    def __init__(self):
        self.organizers = {
            'by_topic': TopicOrganizer(),
            'by_error_type': ErrorTypeOrganizer(),
            'by_difficulty': DifficultyOrganizer(),
            'by_priority': PriorityOrganizer()
        }
        
    def generate_error_book(self, student_errors, organization_method='by_topic'):
        """生成错题本"""
        organizer = self.organizers.get(organization_method, self.organizers['by_topic'])
        
        # 1. 错题分类组织
        organized_errors = organizer.organize(student_errors)
        
        # 2. 添加详细解析
        enriched_errors = self.enrich_with_explanations(organized_errors)
        
        # 3. 生成学习资源链接
        errors_with_resources = self.link_learning_resources(enriched_errors)
        
        # 4. 设计复习计划
        review_plan = self.create_review_plan(errors_with_resources)
        
        # 5. 生成错题本文档
        error_book = self.compile_error_book(
            errors_with_resources, 
            review_plan
        )
        
        return error_book
    
    def create_review_plan(self, errors, algorithm='spaced_repetition'):
        """基于遗忘曲线创建复习计划"""
        if algorithm == 'spaced_repetition':
            plan = SpacedRepetitionScheduler().schedule(errors)
        elif algorithm == 'adaptive':
            plan = AdaptiveScheduler().schedule(errors)
        else:
            plan = FixedIntervalScheduler().schedule(errors)
        
        return {
            'schedule': plan['intervals'],
            'estimated_time': plan['total_time'],
            'success_criteria': self.define_success_criteria(errors),
            'adjustment_rules': self.get_adjustment_rules()
        }

与现有Skill集成

集成calculus-learning-anal