An Introduction to The Common Core State Standards for Mathematics

1. An Introduction to The Common Core State Standards for Mathematics Presented at the Hawaii Department of Education Common Core State Standards Training Sessions January – March 2011 Dewey Gottlieb Educational Specialist for Mathematics Office of Curriculum, Instruction and Student Support

2. Desired Outcomes • Increased understanding of the • CCSS domain progressions • Standards for Mathematical Practice • CCSS-HCPS III Crosswalk documents

3. A Shift in Perspective The CCSS for Mathematics compel a change in the culture of traditional mathematics classroom. In the typical mathematics classroom students are “too busy covering content” to be engaged with mathematics.

4. CCSS for Mathematics • The emphasis on teaching and learning • The CCSS attempts to tell teachers when to slow down and emphasize student understanding of significant mathematical ideas. • “To say, • ‘It was a good lesson but the students didn’t get it’, • is like saying, • ‘The operation was a success, but the patient died.’” • (Lewis, 2002)

5. But what does “higher standards” mean? • More topics? • No. The U.S. curriculum is already cluttered with too many topics • Teaching topics in earlier grades? • No. Analyses of the standards of high-performing countries suggest otherwise. • In Singapore, division of fractions is a 6th grade expectation; in the U.S. it is typically a 4th or 5th grade expectation. • In Japan, probability is introduced in the 7th grade; in the U.S., it can be found anywhere throughout grades 3-6, depending on the state. Standards are “high” for what students take away from cumulative learning experiences

6. A Shift in Perspective Current U.S. curricula (“mile wide, inch deep”) coupled with high-stakes testing pressures teachers to • “cover” at “pace” • turn the page regardless of student needs However, the study of mathematics should not be reduced to merely “a list of topics to cover” • Singapore preaches, “Teach less, learn more”

7. The Domains in the CCSS • Groups of related standards are organized into domains. Domains are overarching big ideas that connect topics across grades. • Standards from different domains may be closely related, conveying an internal coherence among the domains. • In HCPS III, the benchmarks were organized into “strands.” With the transition to CCSS, what we used to call a strand will now be referred to as a domain.

8. Grades K – 5 Mathematics

9. Grades 6 – 8 Mathematics

10. High School Mathematics

11. The Clusters in the CCSS • Within a domain, smaller groups of related standards are organized into clusters. • The clusters help to inform teachers’ decision-making regarding their instructional design and the learning and assessment opportunities provided to students in the mathematics classroom.

12. The Clusters in the CCSS For example, in grade 4, the standards in the Fractions domain are organized into three clusters: • Extend understanding of fraction equivalence and ordering. • Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. • Understand decimal notation for fractions, and compare decimal fractions.

13. The Clusters in the CCSS For example, in grade 6, the standards in “The Number System” domain are organized into three clusters: • Apply and extend previous understandings of multiplication and division to divide fractions by fractions. • Compute fluently with multi-digit numbers and find common factors and multiples. • Apply and extend previous understandings of numbers to the system of rational numbers.

14. The Clusters in the CCSS • We don’t want to simply teach to the standards (i.e., as if checking off a to-do list). • Rather, we want to teach THROUGH the standards, using the specific learning expectations (i.e., the standards) as building blocks for student understanding of significant mathematical ideas (i.e., the clusters) that will prepare them for the mathematics they will be engaging with in subsequent grades.

15. Getting to the Clusters: Teaching THROUGH the Standards Grade 2 Cluster: Use place value understanding and properties of operations to add and subtract. • 2.NBT.5: Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. • 2.NBT.6: Add up to four two-digit numbers using strategies based on place value and properties of operations. • 2.NBT.7: Add and subtract within 1000, using … strategies based on place value, properties of operations, …. • 2.NBT.8: Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900. • 2.NBT.9: Explain why addition and subtraction strategies work, ….

16. Getting to the Clusters: Teaching THROUGH the Standards Grade 6 Cluster: Apply and extend previous understandings of numbers to the system of rational numbers. • 6.NS.5: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values … • 6.NS.6: Understand a rational number as a point on the number line … • 6.NS.7: Understand ordering and absolute value of rational numbers … • 6.NS.8: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane …

17. Learning Progressions: Developing Expertise The brain is a sense-making machine ... it does not store what doesn’t make sense. If we want to make information meaningful to students, we have two options: • Find the prior experience they’ve had and hook the new information to it. OR • Create the experience with them (i.e., build a new network).

18. CCSS for Mathematics The “understand” standards The “understand” standards interact with the “skills” standards to support the development of expertise Students who understand a concept can • Explain it • Demonstrate or illustrate it • Use it into their own arguments and critique someone else’s explanation of it • Show an example of how to apply it (make connections to other mathematical ideas and/or to real-world contexts)

19. Domain Progressions Small Group Task: • Select one domain that goes across grades 6-8. • Review the clusters and standards for that domain for each grade level. • Discuss how the clusters and standards are organized into learning progressions that develop student expertise over time.

20. A Shift in Perspective Too often, students view mathematics as a trivial exercise because they are rarely given the opportunity to grapple with and come to appreciate the intrinsic complexity of the mathematics. Despite our instincts and best intentions, we need to stop “GPS-ing” our students to death. Source: Shannon, A. (2010). Common Core: Two Perspectives on Tasks and Assessments. Presentation at the Urban Mathematics Leadership Network Retreat, June 2010.

21. The Standards for Mathematical Practice “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important processes and proficiencies with longstanding importance in mathematics education.” (CCSS, 2010)

22. The Standards for Mathematical Practice Source: Briars, D. & Mitchell, S. (2010). Getting Started with the Common Core State Standards. A National Council of Supervisors of Mathematics webinar. November 2010.

23. The Standards for Mathematical Practice Conceptual Understanding Strategic Competence Productive Disposition Adaptive Reasoning Procedural Fluency Source: Briars, D. & Mitchell, S. (2010). Getting Started with the Common Core State Standards. A National Council of Supervisors of Mathematics webinar. November 2010.

24. The Standards for Mathematical Practice “Encouraging these practices in students of all ages should be as much a goal of the mathematics curriculum as the learning of specific content” (CCSS, 2010). Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning 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.

25. The Standards for Mathematical Practice The description of each Mathematical Practice begins with the same first three words: Mathematically proficient students … Source: Briars, D. & Mitchell, S. (2010). Getting Started with the Common Core State Standards. A National Council of Supervisors of Mathematics webinar. November 2010.

26. The Standards for Mathematical Practice The Mathematical Practices “describe the thinking processes, habits of mind and dispositions that students need to develop a deep, flexible, and enduring understanding of mathematics; in this sense they are also a means to an end.” Source: Briars, D. & Mitchell, S. (2010). Getting Started with the Common Core State Standards. A National Council of Supervisors of Mathematics webinar. November 2010.

27. The Standards for Mathematical Practice MP #1: Make sense of problems and persevere in solving them. Mathematically proficient students … analyze givens, constraints, relationships and goals … they monitor and evaluate their progress and change course if necessary … and continually ask themselves, “Does this make sense?” Source: Briars, D. & Mitchell, S. (2010). Getting Started with the Common Core State Standards. A National Council of Supervisors of Mathematics webinar. November 2010.

28. Points of Intersection: Content and Practices MP #3: Construct viable arguments and critique the reasoning of others Consider the following subtraction algorithm: • How could I demonstrate the idea that the algorithm always works? 400 – 139  399 – 138 43 – 17  46 – 20

29. Points of Intersection: Content and Practices MP #7: Look for and make use of structure Partitioning • 8 x 7 • 33 + 58

30. Points of Intersection: Content and Practices MP #7: Look for and make use of structure Example: Understanding and interpreting the equation of a line expressed in “Point-Slope Form” y – y1 = m(x – x1)

31. Points of Intersection: Content and Practices MP #4: Model with Mathematics • Model Drawing (“Singapore Math”) • Mrs. Obama has 28 students in her fifth grade class and 2/7 of the class is girls. How many of her students are boys?

32. Points of Intersection: Content and Practices 0% 100% 0 sixth graders 370 sixth graders MP #4: Model with mathematics • Double number lines • Today 40% of the 370 sixth graders at Barack Obama Middle School are on a field trip.

33. Points of Intersection: Content and Practices MP #4: Model with mathematics MP #5: Use appropriate tools strategically • Compare and contrast directly and inversely proportional relationships

34. Points of Intersection: Content and Practices MP #2: Reason abstractly and quantitatively. Consider : • x2 – 1 = (x + 1)(x – 1) • (a + b)2 = a2 + 2ab + b2

35. The Standards for Mathematical Practice Small Group Task: • Select two of the Standards for Mathematical Practice • Identify apparent “points of intersection” between the content standards (in CCSS) and the Standards for Mathematical Practice.

36. CCSS-HCPS III Crosswalks Crosswalk Documents posted at http://standardstoolkit.k12.hi.us/index.html (Click on the “Document Library” link near the top of the webpage)

37. Crosswalk Documents posted at http://standardstoolkit.k12.hi.us/index.html (Click on the “Document Library” link near the top of the webpage)

38. CCSS-HCPS III Crosswalks • Grade level overview • Mapping of CCSS to HCPS III benchmarks • Standards matched to benchmarks • Degree of Match • Comments • Mapping of HCPS III benchmarks to CCSS

39. CCSS-HCPS III Crosswalks Small Group Task: • Gather in groups according to your grade level of interest • Analyze the crosswalk document for your selected grade level to respond to the prompts in the handout

40. Final Thoughts "These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms. It is time to recognize that these standards are not just promises to our children, but promises we intend to keep."

41. A Shift in Perspective • Video: “Math Class Needs a Makeover” • Dan Meyer (a high school mathematics teacher) http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html • After the video: “Think-Pair-Share” • One idea that resonated with you