1. Journey to the Core Focus, 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Journey to the Core

3. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Progression Understanding Focus Coherence

4. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Shared, the same for everyone Essential, fundamental knowledge and skills necessary for student success Adopted and maintained by States; not a federal policy Benchmarks of what students are expected to learn in a content area Common Core State Standards

5. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Great Moderate Strong Magnitude Major Small Minor Not Felt How much of a shift is the Math Common Core for … District School Curriculum Teaching Students

7. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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 K-8 Grade Levels HS Conceptual Categories Standards for Mathematical Practice Standards for Mathematics Content Standards Domains Clusters

10. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee

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

12. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Reflecting…

13. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Reflecting…

14. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Reflecting…

15. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Which is larger? or 3434 6767 Find a common numerator! 6868 6767 Rename or

16. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Focus and Coherence

17. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee CCSS “design principles” Focus Coherence

18. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee The Hunt Institute Video Series Common Core State Standards: A New Foundation for Student Success www.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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Features of Focus and Coherence “Give more detail than teachers were used to seeing in standards.” Fewer Topics Progressions More Detail Show how ideas fit with subsequent or previous grade levels. “Free up time” to do fewer things more deeply. Discuss

20. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Unifying ThemesDetails DomainsClustersStandards

21. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee GradeDomainsClustersStandards K5922 141121 241026 351125 451228 551126 651029 75924 851028 Unifying ThemesDetails GradeDomainsClustersStandards

22. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Conceptual Category DomainsClustersStandards All Standards Advanced Number & Quantity Algebra Functions Geometry Statistics & Probability Modeling Unifying ThemesDetails

23. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Conceptual Category DomainsClustersStandards All Standards Advanced Number & Quantity 49918 Algebra 411234 Functions 410226 Geometry 615376 Statistics & Probability 49229 Modeling **** Unifying ThemesDetails

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

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

26. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee

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

28. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee “Ways of Knowing” the mathematics

29. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee The Hunt Institute Video Series Common Core State Standards: A New Foundation for Student Success www.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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee The number strand “has often been a single strand in elementary school, but in CCSS it is three domains.” Operations and Algebraic Thinking (OA) Number and Operations – Fractions (NF) Number and Operations in Base Ten (NBT) K 1 2 3 4 5 Algebra High School Expressions and Equations (EE) Number System (NS ) Number System (NS ) 6 7 8

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

32. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Properties of the Operations Contextual Situations Contextual Situations Meanings of the Operations The foundation for algebra!

33. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 72 – 29 = ? Mental Math Solve in your head. No pencil or paper! 24 x 25 = ? Nor calculators, cell phones computers, or iPads or....

34. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 72 – 29 = ?24 x 25 = ? Turn and share your reasoning. D iscuss how you: “Decomposed and composed the quantities.” (a.k.a. properties of the operations)

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

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

37. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee The properties of operations. Associative property of addition(a + b) + c = a + (b + c) Commutative property of additiona + b = b + a Additive identity property of 0a + 0 = 0 + a = a Existence of additive inversesFor every a there exists –a so that a + (–a) = (–a) + a = 0 Associative property of multiplication(a × b) × c = a × (b × c) Commutative property of multiplication a × b = b × a Multiplicative identity property of 1a × 1 = 1 × a = a Existence of multiplicative inversesFor every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1 Distributive property of multiplication over addition a × (b + c) = a × b + a × c Not just learning them, but learning to use them.

38. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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 numbers using strategies based on place value and the properties of operations. 5NBT6. Find whole-number quotients and remainders with … using strategies based on place value, properties of operations …. 5NBT7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations….

41. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Develop and use strategies based on properties of the operations

43. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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)

49. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee 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. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Strategies first! Develop and use strategies for learning basic facts before any expectation of knowing facts from memory.

51. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Strategies first! Develop and use strategies to compute with whole numbers, fractions, decimals …. before use of standard algorithms.

52. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Properties of the Operations Contextual Situations Contextual Situations Meanings of the Operations The foundation for algebra!

53. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee In Grades K-8, how many standards reference “real-world contexts” or “word problems”? 54 standards Grade K: OA Grade 1: OA Grade 2: OA, MD Grade 3: OA, MD Grade 4: OA, NF, MD Grade 5: NF, MD, G Grade 6: RP, EE, NS, G Grade 7: RP, EE, NS, G Grade 8: EE, G 24% of K-8 standards

54. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Lots of real-world contexts! Proficient students make sense of quantities and their relationships in problem situations. (MP2) decontexualize & contextualize Lots of real-world contexts! Proficient students make sense of quantities and their relationships in problem situations. (MP2) decontexualize & contextualize

55. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Properties of the Operations Algorithms Real-world Contexts Strategies Walk Away Message Walk Away Message

56. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Great Moderate Major Small Minor Not Felt Strong Shifts in Classroom Practice

57. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Shifts... Content Less data analysis and probability in K-5  More statistics in 6-8 and lots more in HS  Much more emphasis on statistical variability Less algebraic patterns in K-5  Much more algebraic thinking in K-5  More algebra in 7-8 and functions in 8 th More geometry in K-HS  Much more transformational geometry in HS. More focus on Ratio and Proportion beginning in 6th  Percents in 6-7, not in K-5

58. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Shifts… Curriculum & Assessment HS standards as “conceptual categories” not courses ….  supports either integrated or traditional approach or new models that synthesize both approaches. Real-world applications, contexts, and problem solving  Strong emphasis on contexts and word problems from K-HS  Use of measurement contexts across domains, especially “linear” and “liquid” contexts  Multi-step Word Problems beginning in Grade 2  Mathematical modeling interwoven throughout HS

59. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Shifts... Teaching Using a “unit fraction” approach  Understand and use unit fraction reasoning and language and expect it of our students Increased emphasis on visual models  Number line model  Area model Strategies and sense-making before algorithms  Strategies based on properties of the operations  Algorithms culminate years of prior work Discrete to continuous quantities

60. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee And so in closing …

61. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Focus: Unifying themes and guidance on “ways of knowing” the mathematics. Coherence: Progressions across grades based on discipline of mathematics and on student learning. Understanding: Deep, genuine understanding of mathematics and ability to use that knowledge in real-world situations.

62. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Please keep digging, there are many more discoveries in the Core to unearth and we know that the work we are all doing is important for Wisconsin students, for their learning and understanding of mathematics, and for their futures.

63. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Dr. DeAnn Huinker University of Wisconsin-Milwaukee huinker@uwm.edu Thank you! Progression Understanding Focus Coherence

64. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Resources

65. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee CCSSM Resources www.dpi.wi.gov/standards/ccss.html www.mmp.uwm.edu  Quick link: CCSS Resources  www.tinyurl.com/CCSSresources commoncoretools.wordpress.com ime.math.arizona.edu/progressions www.youtube.com/user/TheHuntInstitute#p www.corestandards.org

66. Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Video Series: William McCallum and Jason Zimba lead writers of the CCSSM (The Hunt Institute) The Mathematics Standards: How They Were Developed and Who Was Involved The Mathematics Standards: Key Changes in Their Evidence The Importance of Coherence in Mathematics The Importance of Focus in Mathematics The Importance of Mathematical Practices Mathematical Practices, Focus and Coherence in the Classroom Whole Numbers to Fractions in Grades 3-6 Operations and Algebraic Thinking The Importance of Mathematics Progressions The Importance of Mathematics Progressions from the Student Perspective Gathering Momentum for Algebra Mathematics Fluency: A Balanced Approach Ratio and Proportion in Grades 6-8 Shifts in Math Practice: The Balance Between Skills and Understanding The Mathematics Standards and the Shifts They Require Helping Teachers: Coherence and Focus High School Math Courses www.youtube.com/user/TheHuntInstitute#p