# Teaching Early Algebra through Example-Based Problem Solving: Insights from Chinese and U.S. Element

Two Fundamental Mathematical Ideas: Early AlgebraThree Instructional Principles: IES RecommendationsWorked ExamplesRepresentationsDeep QuestionsTEPS: Teaching through Example-based Problem SolvingNotesReferencesInverse Relations between Addition and SubtractionA Glimpse of Student WorkKnowledge in BriefTypes of Inverse TasksStructure of Additive ProblemsTeaching Additive Inverse through a Part-Whole ModelInsights from Chinese LessonsA Close Look at a Chinese LessonInsights from U.S. LessonsSummary: Teaching Additive Inverse (Part-Whole Model) through TEPSTeaching Additive Inverse through a Comparison ModelInsights from Chinese LessonsInsights from U.S. LessonsBack to the “Jump” Problem from the Opening ScenarioSummary: Teaching Additive Inverse (Comparison Model) through TEPSNotesReferencesInverse Relations between Multiplication and DivisionA Glimpse of Student WorkKnowledge in BriefThe Meaning of MultiplicationStructure of Multiplication ProblemsTypes of Multiplicative InversesTeaching Multiplicative Inverse through an Equal-groups ModelInsights from Chinese LessonsInsights from U.S. LessonsSummary: Teaching Multiplicative Inverse (Equal-Groups Model) through TEPSTeaching Multiplicative Inverse through an Array ModelInsights from U.S. LessonsPossible Integration of the General Chinese ApproachSummary: Teaching Multiplicative Inverse (Array Model) through TEPSTeaching Multiplicative Inverse through a Comparison ModelInsights from Chinese LessonsDeep Questioning about “Times” When Reviewing MultiplicationDeep Questioning on Manipulatives When Teaching DivisionDeep Questioning on Number Line Diagrams When Teaching DivisionInsights from U.S. LessonsSummary: Teaching Multiplicative Inverse (Comparison Model) through TEPSNotesReferencesProperties of Addition: CP and APA Glimpse of Student WorkThe CP of AdditionThe AP of AdditionSummaryKnowledge in BriefDifference between CP and AP of AdditionExpectations from the Common CoreCultural Differences in “When” to TeachTeaching the Commutative Property of AdditionInsights from Chinese LessonsInformal Teaching of the CP of Addition in G1 and G2Formal Teaching of the CP of Addition in G4Insights from U.S. LessonsFormal Teaching of the CP of Addition in G1-G3Informal Teaching of the CP of Addition in G1 and G2Additional ObservationsSummary: Teaching the CP of Addition through TEPSTeaching the Associative Property of AdditionInformal Teaching of AP of AdditionInsights from Chinese LessonsInsights from U.S. LessonsFormal Teaching of AP of AdditionInsights from Chinese LessonsInsights from U.S. Lessons: G3Summary: Teaching the AP of Addition through TEPSNotesReferencesProperties of Multiplication: CP, AP, and DPA Glimpse of Student WorkStudent Response to the CP (×) ItemsStudent Response to the AP (×) ItemsStudent Response to the DP itemsKnowledge in BriefCP, AP, and DPExpectations from the Common CoreModeling the PropertiesCultural Differences in “How” and “When”Teaching the Commutative Property of MultiplicationInsights from Chinese LessonsInformal Teaching of the CP (×)Formal Teaching of the CP (×)Insights from U.S. LessonsInformal Teaching of the CP(×)Formal Teaching of CP(×)Summary: Teaching the CP of Multiplication through TEPSTeaching the Associative Property of MultiplicationInsights from Problem-Solving Lessons: Illustrating the APChinese Grade 3 Lessons: Informal TeachingChina G4 Lesson: Formal TeachingU.S. G3 and G4 lessons: Formal TeachingInsights from Computational Strategy Lessons: Applying the APChina G3 lesson: Informal TeachingU.S. G4 Lessons: Formal and Informal TeachingSummary: Teaching the AP of Multiplication through TEPSTeaching the Distributive PropertyInsights from Chinese lessonsInformal Teaching of the DPFormal Teaching of the DPInsights from the U.S. LessonThe Array/Area Model: Breaking Apart in Different WaysAdding Story Contexts for the Array ModelSummary: Teaching the DP through TEPSNotesReferencesTEPS: The Teaching of Early AlgebraSituating a Worked Example in a Real-world ContextModeling the Real-World Context with Concreteness FadingAsking Concept-specific Questions to Promote Meaning-makingAsking Comparison Questions to Promote Connection-makingBeyond Teaching Early Algebra: More ImplicationsFuture DirectionsNotesReferencesAppendix A: Example Chinese G1 Lesson Plan about Fact Families (Additive Inverse)Addition with the Sum of 8 and Subtraction from 8I Introduction to Activate InterestsII New LessonSelf-exploration of the new knowledgeReinforcement, Application, and ExtensionIII Summary of the LessonAppendix B: Example Chinese G1 Lesson Plan about “Making a Ten to Add” (Informal Teaching of the AP)Plus SomethingTeaching ProcessI Create a Scenario, and Introduce the Lesson with ReviewIII Find the Pattern and Enhance ThinkingIV SummaryNotesAppendix C: Example Chinese G4 Lesson Plan about Formal Teaching of the CP and AP of MultiplicationCommutative and Associative Properties of MultiplicationI Introduction with ConversationII Communication and SharingIII Feedback and ImprovementIV Reflection and SummaryNotes