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Math PracticeStudentsTeachers1. Make sense of problems and persevere in solving them.Have an understanding of the situationAnalyze information in problems (givens, constrains, relationships, goals) Make conjectures and plan a solution pathway Turn and talk for first steps or generate a solution plan.Use patience and persistence to solve problemBe able to use multiple strategiesMonitor and evaluate the progress and change course as necessary Use self-evaluation and redirectionsCommunicate both verbally and writtenShow patience and listen to others.Be able to deduce what is a reasonable solutionAssess the reasonableness of process.Assess the reasonableness of an answer and ask, “Does this make sense?”Comments:Provide open-ended and rich problemsAsk probing questions Model multiple problem-solving strategies through Think- AloudPromotes and values discourseCross-curricular integrationsPromotes collaborationProbe student responses (correct or incorrect) for understanding Model and accept multiple approachesProvide scaffolding appropriatelyProvide a safe environment for learning from mistakesComments:2. Reason abstractly and quantitatively.Create and use multiple representationsInterpret problems in contextsEstimate first/answer reasonableMake connectionsVisualize problemsRepresent abstract and contextual situations symbolically.Put symbolic problems into context.Talk about problems, real life situationsUnderstand the meaning of quantities Attend to unitsMake sense of quantities and relationships in problem situations Create a coherent representation of the problem at hand Flexibly use properties of operationsComments:Develop opportunities for and model problem solving strategiesGive time for processing and discussingTie content areas together to help make connectionsGive real world situationsThink aloud for student benefitValue invented strategies and representationsLess emphasis on the answerModel context to symbol and symbol to context.Create problems such as, “What word problem will this equation solve?”Offer authentic performance ments:Math PracticeStudentsTeachers3. Construct viable arguments and critique the reasoning of others.Ask questionsUse examples and counter examples to build a logical progression of statements to explore and support their ideasReason inductively and make plausible argumentsUse definitions and previously established causes/effects (results) in constructing arguments Use objects, drawings, diagrams, and actions to communicate and defend mathematical reasoningDevelop ideas about mathematics and support their reasoning with mathematical evidence.Decide if the arguments of others make sense and ask probing questions to clarify or improve the arguments Respond to the arguments of others well. Use of precise mathematics vocabularyComments:Create a safe environment for risk-taking and critiquing with respectModel respectful discourse behaviors.Provide complex, rigorous tasks that foster deep thinkingProvide find-the-error problems.Provide time for student discoursePlan effective questions and student groupingProbe studentsComments:4. Model with mathematics.Realize they use mathematics (numbers and symbols) to solve/work out real-life situationsSymbolize real-world problems with math.Choose and apply representations, manipulatives, and other models to solve problems.Apply prior knowledge to solve problems.Use strategies to make problems simpler.Analyze relationships to draw conclusionsInterpret mathematical results in contextUse estimation and logic to check the reasonableness of an answer.Make sense of the mathematicsComments:Allow time for the process to take place (model, make graphs, etc.)Create an emotionally safe environment where risk taking is valuedModel desired behaviors (think alouds) and reasoning skills (questioning, revision, reflection/written)Model various modeling techniques.Make appropriate tools availableProvide meaningful, real world, authentic, performance-based tasks (nontraditional work problems)Accept and value multiple approaches and ments:Math PracticeStudentsTeachers5. Use appropriate tools strategically.Choose the appropriate tool to solve a given problem and deepen their conceptual understanding (paper/pencil, ruler, base 10 blocks, compass, protractor)Choose the appropriate technological tool to solve a given problem and deepen their conceptual understanding (e.g., spreadsheet, geometry software, calculator, web 2.0 tools)Identify and locate resources.Identify relevant external math resources (digital content on a website) and use them to pose or solve problems Compare the efficiency of different toolsDefend mathematically the choice of a tool.Recognize the usefulness and limitations of different toolsComments:Maintain knowledge of appropriate toolsProvide a toolbox at all times with all available tools; students then choose as needed.Effective modeling of the tools available, their benefits and limitationsModel a situation where the decision needs to be made as to which tool should be usedCompare/contrast effectiveness of toolsMake available and encourage use of a variety of toolsComments:6. Attend to municate (orally and in writing) with precise vocabulary.Use mathematics concepts and vocabulary appropriatelyState meaning of symbols and use appropriatelyAttend to units/labeling/tools accuratelyLabel accurately when measuring and graphing Carefully formulate explanations and defend answersFormulate and make use of definitions with others and their own reasoningCalculate accurately and efficiently, expressing numerical answers with a degree of precisionExpress answers within context when appropriate Ensure reasonableness of answersPerseverance through multiple-step problemsComments:Encourage students to think aloud/talk aloudExplicit instruction/teacher model of think aloud/talk aloudGuided Inquiry including teacher gives problem, students work together to solve problems, and debriefing time for sharing and comparing strategiesModel problem-solving strategies.Give explicit and precise instruction.Probing questions targeting content of studyPromote mathematical lingoGive room to discuss why wrong answers are wrongUse English language arts strategies of decoding, comprehending, and text to-self connections for interpreting symbolic and contextual math ments:Math PracticeStudentsTeachers7. Look for and make use of structure.Look for, interpret, and identify patterns and structuresMake connections to skills and strategies previously learned to solve new problems/tasks independently and with peersReflect and recognize various structures in mathematicsBreakdown complex problems into simpler, more manageable chunksBe able to “step back” / shift perspectiveUse multiple representations for quantities.View complicated quantities as both a single object and a composition of objects.Value multiple perspectivesRecognize the significance in concepts and models and use the patterns or structure for solving related problems Comments:Be quiet and structure opportunities for students to think aloudFacilitate learning by using open-ended questioning to assist students in explorationCareful selection of tasks that allow for students to discern structures or patterns to make connectionsAllow time for student discussion and processing in place of fixed rules or definitionsFoster persistence/stamina in problem solvingThrough practice and modeling time for studentsAsk for multiple interpretations of quantities.Use open-ended ments:8. Look for and express regularity in repeated reasoning.Identify patterns and make generalizationsNotice repeated calculations and look for general methods and shortcuts Generate rules from repeated reasoning or practice (e.g., integeroperations).Continually evaluate reasonableness of intermediate results (comparing estimates)Evaluate the reasonableness of intermediate steps.Maintain oversight of the processComments:Provide rich and varied tasks that allow students to generalize relationships and methods, and build on prior mathematical knowledgeDon’t teach steps or rules, but allow students to explore and generalize to discover and formalize.Provide adequate time for explorationProvide time for dialogue and reflection, peer collaborationAsk deliberate questions that enable students to reflect on their own thinkingCreate strategic and intentional check in points during student work timeComments: ................
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