Mathematical Practice: Construct viable arguments and critique the reasoning of others. Build logical progression of statements to explore conjectures Recognize and use counterexamples Justify their conclusions and respond to others ideas using drawings, diagrams, actions 25
Mathematical Practice: Model with mathematics. Solve problems in everyday life Write equation to describe a situation Solve a design problem Make assumptions and approximations to simplify a complicated situation Interpret results to see if they make sense in terms of the situation 30
Singing at the Ballpark- 4th 21 students need to get to the ballpark. Each car will carry one adult and up to 4 students. 5 ¼ cars please? 31
Interpret results into context to check for reasonableness 32
Mathematical Practice: Use appropriate tools strategically. Use paper pencil, concrete models, ruler, protractor, calculator, spreadsheet, etc. Tools might also include choosing an appropriate mathematical strategy 39
Tools Calculator – Strategy Convert to decimals 42
Mathematical Practice: Attend to precision. Communicate precisely with clear definitions State meaning of symbols Calculate accurately, efficiently, and use appropriate level of precision 43
Have a Clear Definition Teacher comments on 2-4 tests: Why do the tasks use the word angle when our textbooks use vertex? 44
Words are tools for thinking Not having words to use limits the mathematics we can think about. – Harold Asturias, Lawrence Hall of Science Teacher comments: Why did you use the worddimensions on the 5 th grade task box of cubes? Why didnt you just ask for length, width, and height? 45
Cognitive Demand How many dimensions are there? 46
Making Sense of Structure Seven most important words to transform education: How did you Figure that out? 52
7 th Grade Freezing in Fargo How many times colder is Wednesday Feb. 25 th than Tuesday Feb. 3 rd ? Almost 40% of the students in the sample subtracted. 53
Mathematical Practice: Look for and express regularity in repeated reasoning. Look for repeated calculations in both general methods and for short cuts 54
6 th Freezing In Fargo Which week (Sunday through Saturday) recorded the average lowest temperature? A student noticing that all the averages are divide by 7 days should realize that comparing totals will yield some comparative results without needing to divide. 55
Tools For Practices and Standards Use MARS Tasks – Define the meaning of the standards and practices – Raise expectations for teachers about what students are capable of accomplishing – Help teachers anticipate misconceptions so that they can be surfaced and addressed in class discussion and re-engagement lessons 56
Resources SVMIMAC.org website Inside Mathematics.org directly or through link in NCSM 57
TOOLS BY SUBJECTAlgebra & FunctionsAlgebraic Properties & RepresentationsData AnalysisFunctions & RelationsGeometry & MeasurementMathematical Reasoning & ProofsNumber OperationsNumber PropertiesPatterns, Functions & AlgebraProbabilityStatisticsAlgebra & FunctionsAlgebraic Properties & RepresentationsData AnalysisFunctions & RelationsGeometry & MeasurementMathematical Reasoning & ProofsNumber OperationsNumber PropertiesPatterns, Functions & AlgebraProbabilityStatistics 3rd GradeCore IdeasRecognize and use characteristics, properties, and relationships of two-dimensional geometric shapes and apply appropriate techniques to determine measurements.Choose appropriate units and tools for particular tasks and use these units and tools to estimate and measure (length, weight, temperature, time, and capacity).Identify and compare attributes of two-dimensional shapes and develop vocabulary to describe the attributes.Calculate perimeter and area and be able to distinguish between the two measures. (Area may be measured by covering a figure with squares.)Use visualization, spatial reasoning, and geometric modeling to solve problems.Recognize geometric ideas and relationships and apply them to problems.MARS TasksLooking Glass LandTaskRubricCore Mathematical Ideas and ChallengesQuestions for Teacher ReflectionDiscussion of Successful Examples of Student WorkDiscussion of Student MisconceptionsGraph and Analysis of the MARS Task DataSummary of Student Understandings and MisunderstandingsImplications for InstructionTaskRubricCore Mathematical Ideas and ChallengesQuestions for Teacher ReflectionDiscussion of Successful Examples of Student WorkDiscussion of Student MisconceptionsGraph and Analysis of the MARS Task DataSummary of Student Understandings and MisunderstandingsImplications for Instruction 58
Practices Require Content Looking at and Understanding Number System Using Place Value Strategies to Make Sense of and Solve Problems Understanding Number Line as a basic mathematical concept and tool
Butterfly and Moth Collection How much longer was the longest wingspan than the shortest?
Research Suggests: Number lines help students understand fractions as a single number instead of two – unique point or location on the line Number line concepts and reading fractions can be introduced through rulers, clocks, scales Number lines help students develop the ability to generalize about number and operations
8 th Grade - Tool Understand the 8 th grade Mathematics Common Core Course and the deep rich mathematical expectations for students Includes rich Algebra Strand but also works to expand and deepen understanding and facility with other strands Use as Placement Test or Summative Test - Could your 7 th graders pass this test? Could your Algebra and Geometry students meet standard on test? Facilitate Course Discussion with Staff and Parents – Is this the mathematics we want students to be able to know and do?
Implementing Formative Assessment Lessons- Forming Professional Learning Communities – How do we create classroom culture? – How do we facilitate staff doing mathematics together to understand purpose of the lesson, understand the mathematics before giving it to students, work as a learning community to discuss techniques, purpose of using student work, types of comments to put on student papers, how to have a plenary discussion? What are the important mathematical ideas to be drawn from students? What teacher moves help hold all students accountable for their own learning? – Making time for lessons?