1. Coherence in the AZCCRS
September 6, 2014

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

3. Ways of Doing?

4. 08/13/09 4
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The Broomsticks 4

5. The Broomsticks 08/13/09 The RED broomstick is three feet long5
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The Broomsticks
The RED broomstick is three feet long
The YELLOW broomstick is four feet long
The GREEN broomstick is six feet long
Source: 5

6. Source: http://tedcoe

7. Source: http://tedcoe

8. Source: http://tedcoe

9. Source: http://tedcoe

10. 08/13/09
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11. Source:

12. Source:

13. Key Shifts in the AZCCRS
Focus
Coherence
Rigor
Source:

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

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

16. 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., pp.48-49

17. 4.OA.1, 4.OA.2 From the CCSS: Grade 4
Interpret a multiplication equation as a comparison, e.g., interpret = 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

18. 4.OA.1, 4.OA.2 From the CCSS: Grade 4
Interpret a multiplication equation as a comparison, e.g., interpret = 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

19. 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

20. “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., p.49

21. What do we mean when we talk about “measurement”?
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What do we mean when we talk about “measurement”?
Measurement

22. But what does he mean by “comparison”?
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“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

23. How about this? 08/13/09 Determine the attribute you want to measure
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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

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

25. The circumference is three and a bit times as large as the diameter.

26. The circumference is about how many times as large as the diameter?
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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

28. 2828
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What is an angle?
Angles 28

29. What attribute are we measuring when we measure angles?
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What attribute are we measuring when we measure angles?
Angles 29

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

32. Source:

34. Similar Figures

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

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

38. Assume

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

40. Materials?
Source:

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

42. 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

43. Higher Ed Impact?
Placement
Assessment
Content
Instruction
Preservice

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