Coherence in the AZCCRS September 6, 2014

Teaching and Learning Mathematics Ways of doing Ways of thinking Habits of thinking

Ways of Doing?

08/13/09 4 08/13/09 The Broomsticks 4

The Broomsticks 08/13/09 The RED broomstick is three feet long 5 08/13/09 08/13/09 The Broomsticks The RED broomstick is three feet long The YELLOW broomstick is four feet long The GREEN broomstick is six feet long Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc 5

Source: http://tedcoe

Source: http://tedcoe

Source: http://tedcoe

Source: http://tedcoe

08/13/09 1010 08/13/09 10

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Key Shifts in the AZCCRS Focus Coherence Rigor Source:

Learning Progressions in the AZCCRS Ways of Thinking? Learning Progressions in the AZCCRS

From the CCSS: Grade 3 Source: CCSS Math Standards, Grade 3, p. 24 (screen capture)

From the CCSS: Grade 3 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. Soucre: CCSS Grade 3. See: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. pp.48-49

4.OA.1, 4.OA.2 From the CCSS: Grade 4 Interpret 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. Source: CCSS Grade 4

4.OA.1, 4.OA.2 From the CCSS: Grade 4 Interpret 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. Source: CCSS Grade 4

From the CCSS: Grade 5 5.NF.5a Interpret 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. Source: 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”? 2121 08/13/09 What do we mean when we talk about “measurement”? Measurement

But what does he mean by “comparison”? 2222 08/13/09 08/13/09 “Technically, 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”? Measurement 22

How about this? 08/13/09 Determine the attribute you want to measure 2323 08/13/09 08/13/09 How about this? Determine the attribute you want to measure Find something else with the same attribute. Use it as the measuring unit. Compare the two: multiplicatively. Measurement 23

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

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

The circumference is about how many times as large as the diameter? 08/13/09 08/13/09 The circumference is about how many times as large as the diameter? The diameter is about how many times as large as the circumference? 26 26

2828 08/13/09 What is an angle? Angles 28

What attribute are we measuring when we measure angles? 2929 08/13/09 What attribute are we measuring when we measure angles? Angles 29

CCSS, Grade 4, p.31 Source: CCSS Grade 4, 4.MD.5

http://tedcoe.com/math/radius-unwrapper-2-0

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Similar Figures

CCSS: Grade 7 (p.46) Source: CCSS Grade 7, p.46

http://tedcoe.com/math/algebra/constant-rate http://tedcoe.com/math/algebra/constant-rate

CCSS: Grade 8 (8.EE.6, p.54) Source: CCSS Grade 8

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

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

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

Teaching and Learning Mathematics Ways of doing Ways of thinking Habits of thinking

Standards for Mathematical Practice Eight Standards for Mathematical Practice Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the understanding of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Source: CCSS

Higher Ed Impact? Placement Assessment Content Instruction Preservice

Ted Coe Director, Mathematics Achieve, Inc. tcoe@achieve.org