Dr. DeAnn Huinker University of Wisconsin-Milwaukee huinker@uwm Wisconsin Mathematics Council

1. Journey to the CoreFocus, Coherence, and Understanding in the Common Core State Standards for Mathematics Dr. DeAnn Huinker University of Wisconsin-Milwaukee huinker@uwm.edu Wisconsin Mathematics Council Green Lake, Wisconsin 4 May 2012

2. Journey to the Core

3. Progression Focus Coherence Understanding Progression Progression

4. Common CoreState Standards Adopted and maintained by States; not a federal policy Shared, thesame for everyone Benchmarks of what students are expected to learn in a content area Essential, fundamental knowledge and skills necessary for student success

5. We are learning to... • Understand “Focus”and “Coherence” • Consider how the standards detail or specify “Ways of Knowing” mathematics • Embrace“Shifts” • content topics • curriculum & assessment • instructional approaches

6. How much of a shift is theMath Common Core for … • District • School • Curriculum • Teaching • Students Great Major Strong Moderate Small Minor Not Felt Magnitude

7. A Long Overdue Shifting of the Foundation For as long as most of us can remember, the K-12 mathematics program in the U.S. has been aptly characterized in many rather uncomplimentary ways: underperforming, incoherent, fragmented, poorly aligned, narrow in focus, skill-based, and, of course, “a mile wide and an inch deep.” ---Steve Leinwand, Principal Research Analyst American Institutes for Research in Washington, D.C

8. But hope and change have arrived! Like the long awaited cavalry, the new Common Core State Standards for Mathematics(CCSS) presents us a once in a lifetime opportunity to rescue ourselves and our students from the myriad curriculum problems we’ve faced for years. ---Steve Leinwand, Principal Research Analyst American Institutes for Research in Washington, D.C Make no mistake, for K-12 math in the United States, this IS a brave new world. --Steve Leinwand

9. Make sense of problems Reason quantitatively Viable arguments & critique Model with mathematics Strategic use of tools Attend to precision Look for and use structure Look for regularity in reasoning Standards for Mathematical Practice Standards for Mathematics Content • K-8 Grade Levels • HS Conceptual Categories Domains Clusters Standards

11. Digging in… Begin to unearth some discoveries: • Mathematics content • Teaching of mathematics • Student “knowing” of mathematics

12. Reflecting… 2NBT9. Explain why addition and subtraction strategies work, using place value and the properties of operations. 3OA3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

13. Reflecting… 4NF2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

14. Reflecting… 4NF2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

15. Find a common numerator! Which is larger? 3 4 6 8 6 7 6 7 or Rename or

16. Focus and Coherence

17. CCSS “design principles” Focus Coherence

18. The Hunt Institute Video SeriesCommon Core State Standards: A New Foundation for Student Successwww.youtube.com/user/TheHuntInstitute#p Helping Teachers: Coherence and Focus Dr. William McCallum Professor of Mathematics, University of Arizona Lead Writer, Common Core Standards for Mathematics

19. Discuss Features of Focus and Coherence Fewer Topics “Free up time” to do fewer things more deeply. “Give more detail than teachers were used to seeing in standards.” Show how ideas fit with subsequent or previous grade levels. Progressions More Detail

20. Unifying Themes Details

21. Unifying Themes Details

22. Unifying Themes Details

23. Unifying Themes Details

24. Content Standards: Reflect hierarchical nature & structure of the discipline. – Progressions – Ways of Knowing Practice Standards: Reflect how knowledge is generated within the discipline. Discipline of mathematics Research on students’ mathematics learning Coherence Reflects what we know about how students develop mathematical knowledge. Reflects the needs of learnersto organize and connect ideas in their minds (e.g., brain research).

25. CCSSM Progression Documents (draft) by The Common Core Standards Writing Team Required Professional Reading & Discussion Comprehensive discussions on:• Intent of specific standards. • Development within and across grades. • Connections across domains. • Suggested instructional approaches. ime.math.arizona.edu/progressions

27. Focus and Coherence Domains and Clusters as unifying themes within & across grades. Detail in the standards give guidance on “ways of knowing” the mathematics Embedded progressionsof mathematical ideas.

28. “Ways of Knowing” the mathematics

29. The Hunt Institute Video SeriesCommon Core State Standards: A New Foundation for Student Successwww.youtube.com/user/TheHuntInstitute#p Operations and Algebraic Thinking Dr. Jason Zimba Professor of Physics and Mathematics Bennington College, Vermont Lead Writer, Common Core Standards for Mathematics

30. The number strand “has often been a single strand in elementary school, but in CCSS it is three domains.” Operations and Algebraic Thinking (OA) Algebra Expressions and Equations (EE) Number and Operations in Base Ten (NBT) Number System (NS) Number and Operations – Fractions (NF) 6 7 8 High School K 1 2 3 4 5

31. Operations & Algebraic Thinking (OA) ‘“Addition, subtraction, multiplication, & division have meanings, mathematical properties, and uses that transcendthe particular sorts of objects that one is operating on, whether those be multi-digit numbers or fractions or variables or variables expressions.”

32. Meanings of the Operations Properties of the Operations Contextual Situations The foundation for algebra!

33. 72 – 29 = ? 24 x 25 = ? Mental Math Solve in your head. No pencil or paper! Nor calculators, cell phones computers, or iPads or ....

34. 72 – 29 = ? 24 x 25 = ? Turn and share your reasoning. Discuss how you: “Decomposed and composed the quantities.” (a.k.a. properties of the operations)

35. 24 x 25 = ? I thought 25 x 25 = 625 and then I subtracted 25. 625 – 25 = 600. I thought 24 x 100 = 2400, and 2400 ÷ 4 = 600. I figured that there are 4 twenty-fives in 100, and there are 6 fours in 24, so 100 x 6 = 600.

36. 24 x 25 = ? 25 x 4 = 100, 6 x 100 = 600, 600 + 100 = 700. “I would try to multiply in my head, but I can't do that.” Well, 10 x 25 = 250, 2(10 x 25) = 500, 500 x 4 = 2000.

37. The properties of operations. Not just learning them, but learning to use them.

38. And in the domain of Operations and Algebraic Thinking, it is those meanings, properties, and uses which are the focus; and it is those meanings, properties, and uses that will remain when students begin doing algebra in middle grades [and beyond]. --Jason Zimba

39. In Grades K-8, how many standards reference “properties of the operations”? 28 standards Grade 1: OA, NBT Grade 2: NBT Grade 3: OA, NBT Grade 4: NBT, NF Grade 5: NBT Grade 6: NS, EE Grade 7: NS, EE Grade 8: NS 12% of K-8 standards

40. Using properties of operations • 1OA3. Apply properties of operations as strategies to add and subtract. • 3OA5. Apply properties of operations as strategies to multiply and divide. • 4NBT5. Multiply two two-digit numbersusing strategies based on place value and the properties of operations. • 5NBT6. Find whole-number quotientsand remainders with … using strategies based on place value, properties of operations…. • 5NBT7. Add, subtract, multiply, and divide decimalsto hundredths, using concrete models or drawings and strategies based on place value, properties of operations….

41. 6EE3. Apply the properties of operations to generate equivalent expressions. • 7NS2c: Apply properties of operations as strategies to multiply and divide rational numbers. • 7EE1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. • and into high school……

42. Develop and use strategies based on properties of the operations

43. CCSS Glossary Computation strategy Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another. Computation algorithm A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly.

44. In Grades K-8, how many standards reference using “strategies”? 26 standards Grade K: CC Grade 1: OA, NBT Grade 2: OA, NBT Grade 3: OA, NBT Grade 4: NBT, NF Grade 5: NBT Grade 7: NS, EE 11% of K-8 standards

45. Standard 1OA6: “Basic Facts” Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

46. Standard 3OA5: Basic Facts Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) Turn around facts Double a known fact Use a helping fact

47. In Grades K-8, how many standards reference using “algorithms”? 5 standards Grade 3: NBT2 Grade 4: NBT4 Grade 5: NBT5 Grade 6: NS2, NS3 2% of K-8 standards

48. Algorithms Grade 3 “use strategies and algorithms” to add and subtract within 1000. (Footnote: A range of algorithmsmay be used.) (3NBT2) Grade 4 “use the standard algorithm” to add and subtract multi-digit whole numbers. (4NBT4) Grade 5 “use the standard algorithm” to multiply multi-digit whole numbers. (5NBT4) Grade 6 “use the standard algorithm” to divide multi-digit numbers and to divide multi-digit decimals. (6NS2, 6NS3)

49. Algorithms Grade 3 “use strategies and algorithms” to add and subtract within 1000. (Footnote: A range of algorithms may be used.) (3NBT2) Grade 4 “use the standard algorithm” to add and subtract multi-digit whole numbers. (4NBT4) Grade 5 “use the standard algorithm” to multiply multi-digit whole numbers. (5NBT4) Grade 6 “use the standard algorithm” to divide multi-digit numbers and to divide multi-digit decimals. (6NS2, 6NS3)

50. Strategies first!Develop and use strategies for learning basic facts before any expectation of knowing facts from memory.