common core and the community college may 20, 2014

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Common Core and the Community College

May 20, 20141Teaching and Learning MathematicsWays of doing Ways of thinkingHabits of thinkingWays of Doing? The Broomsticks

08/13/09408/13/094

The RED broomstick is three feet longThe YELLOW broomstick is four feet longThe GREEN broomstick is six feet long

The BroomsticksSource: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

08/13/09508/13/0956

Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

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9Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

08/13/091008/13/0910Ways of Thinking?Learning Progressions in the Common CoreFrom the CCSS: Grade 312Source: CCSS Math Standards, Grade 3, p. 24 (screen capture)

3.OA.1:Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 7.From the CCSS: Grade 313Soucre: CCSS Grade 3. See: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. pp.48-494.OA.1, 4.OA.2Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

From the CCSS: Grade 414Source: CCSS Grade 4

4.OA.1, 4.OA.2Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

From the CCSS: Grade 415Source: CCSS Grade 4

5.NF.5aInterpret multiplication as scaling (resizing), by:Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.From the CCSS: Grade 516Source: CCSS Grade 5

In Grades 6 and 7, rate, proportional relationships and linearity build upon this scalar extension of multiplication. Students who engage these concepts with the unextended version of multiplication (a groups of b things) will have prior knowledge that does not support the required mathematical coherences.Source: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. p.49

What do we mean when we talk about measurement?Measurement08/13/0918Technically, a measurement is a number that indicates a comparison between the attribute of an object being measured and the same attribute of a given unit of measure.Van de Walle (2001)But what does he mean by comparison?Measurement08/13/091908/13/0919How about this?Determine the attribute you want to measureFind something else with the same attribute. Use it as the measuring unit.Compare the two: multiplicatively.Measurement08/13/092008/13/0920

Source: Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)

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The circumference is three and a bit times as large as the diameter.http://tedcoe.com/math/circumference23

Similar Figureshttp://tedcoe.com/math/algebra/constant-rate

http://tedcoe.com/math/algebra/constant-rateCCSS: Grade 8 (8.EE.6, p.54)26Source: CCSS Grade 8

Assume

http://tedcoe.com/math/geometry/similar-triangles

CCSS: Geometry (G-SRT.6, p. 77)28Source: CCSS High School Geometry (screen capture)

Teaching and Learning MathematicsWays of doing Ways of thinkingHabits of thinkingStandards for Mathematical Practice30Eight Standards for Mathematical PracticeMake sense of problems and persevere in solving themReason abstractly and quantitativelyConstruct viable arguments and critique the understanding of othersModel with mathematicsUse appropriate tools strategicallyAttend to precisionLook for and make use of structureLook for and express regularity in repeated reasoningSource: CCSS30Assessment?31The error of excessively literal readingReading individual standards as individual ingredients of a test. When the explicit goal is to have the ingredients cook into a cake, tasting the uncooked ingredients is a poor measure of how the cake tastes (although it is related). The goal, as stated in the grade-level introductions and the practices standards, is for the students to cook.Source: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. p.50-51

Grade 5:

AIMS (Arizona) Sample Test33

Source: http://www.azed.gov/standards-development-assessment/aims/sample-for-students/ (April 2014)

Grade 5:

AIMS (Arizona) Sample Test34Source: http://www.azed.gov/standards-development-assessment/aims/sample-for-students/ (April 2014)

Grade 6:AIMS (Arizona) Sample Test35

Around 96oSource: http://www.azed.gov/standards-development-assessment/aims/sample-for-students/ (April 2014)

High School:AIMS (Arizona) Sample Test36

Source: http://www.azed.gov/standards-development-assessment/aims/sample-for-students/ (April 2014)

Standard: Solve square root equations involving only one radical.36Next Generation Mathematics Assessments: Beyond doing37Next-Gen Assessment Claims

Smarter Balanced Claims

http://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/09/Smarter-Balanced-Mathematics-Claims.pdfSmarter Balanced Claims

http://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/09/Smarter-Balanced-Mathematics-Claims.pdfPARCC Mathematics Claims

Source: PARCC

41Standards for Mathematical Practice42Eight Standards for Mathematical PracticeMake sense of problems and persevere in solving themReason abstractly and quantitativelyConstruct viable arguments and critique the understanding of othersModel with mathematicsUse appropriate tools strategicallyAttend to precisionLook for and make use of structureLook for and express regularity in repeated reasoningSource: CCSSThe mathematical practices are intended to foster reasoning and sense making. 42Type I TasksType II TasksType III TasksPARCC Design4343Type I Tasks:Concepts, skills and procedures

Balance of conceptual understanding, fluency, and applicationCan involve any or all mathematical practice standardsMachine scorable including innovative, computer-based formatsSub-claims A, B and E

PARCC Design44Source: PARCC

44Type II Tasks:Expressing mathematical reasoning

Each task calls for written arguments / justifications, critique of reasoning, or precision in mathematical statements (MP.3, 6). Can involve other mathematical practice standardsMay include a mix of machine scored and hand scored responsesSub-claim C


45Type III Tasks:Modeling / applicationsEach task calls for modeling/application in a real-world context or scenario (MP.4) Can involve other mathematical practice standardsMay include a mix of machine scored and hand scored responsesSub-claim D


46PARCC Sample Items47

Source: http://www.parcconline.org/samples/mathPARCC Sample Items48Source: http://www.parcconline.org/samples/math

PARCC Sample Items49Source: http://www.parcconline.org/samples/math

50Source:

Source: http://www.parcconline.org/samples/mathPARCC Sample Items51

Source: http://www.parcconline.org/samples/mathPARCC Sample ItemsPARCC Sample Items52Source: http://www.parcconline.org/samples/math





Algebra IPARCC Sample Items57Source: http://www.parcconline.org/samples/math



Materials?60Source: http://ime.math.arizona.edu/progressions/

Materials?61Source: EngageNY

Materials?62Source: EngageNY

EQuIP63Source: www.achieve.org/equip

Community College Impact?PlacementAssessmentContentInstructionPreservice

Ted CoeDirector of MathematicsAchieve, [email protected]

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common core and the community college may 20, 2014

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