CGCS Mathematics Mathematics Retreat, September 21-22, 2011 –Jason Zimba, lead writer of the CCSS Mathematics Mathematics Advisory Committee professional development –Mathematics Learning Progressions, March 19-20, 2012 –Phil Daro, lead writer of the CCSS Mathematics Mathematics Retreat –Learning Progressions, June 19, 2012 –William McCallum, lead writer of the CCSS Mathematics Pre-conference in Mathematics –July 11, 2012 –William McCallum & IM&E
CGCS: Mathematics September 21-22, 2011 – Albany, New York Audience: District mathematics leaders Purpose: Develop a shared understanding of the Common Core Mathematics Standards and examine assessment items that probe for deeper conceptual understanding Facilitators: Student Achievement Partners (including Jason Zimba, lead developer of the Common Core Mathematics Standards)
Common Core Mathematics Standards Umbrella Balanced approach to mathematics –Three instructional shifts that correspond to the design principles underlying the development of the standards Focus Coherence Rigor: deep understanding, fluency, and applications
The Importance of Focus How the curriculum in the U.S. was organized
K 12 Number and Operations Measurement and Geometry Algebra and Functions Statistics and Probability Traditional U.S. Approach
Focusing attention within Number and Operations Operations and Algebraic Thinking Expressions and Equations Algebra Number and Operations— Base Ten The Number System Number and Operations— Fractions K12345678High School
Focus in mathematics: The new version Everyone’s Time and Effort
Why Focus? Provides the time for students to transfer mathematical skills and understanding across concepts and grade levels –Mathematical connections –Deep conceptual understanding –Connect conceptual and procedural understanding Transitions from concrete↔pictorial↔language↔abstract Deepens and narrows the scope of how time and energy is spent in the math classroom –Communicate focus so that instruction is manageable; it is more than merely writing a standard a day
Focus– strengthens foundations Progression involving fractional concepts (conceptual understanding) and operations (multiplication and division of fractions): –Grade Three: Develop understanding of fractions as numbers Develop understanding of fractions as part of a whole and as a number on the number line Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size –Grade Four: Extend understanding of fraction equivalence and ordering Build fractions from unit fractions by applying and extending previous understandings on whole numbers (decompose a fraction into a sum of fractions with the same denominator)
Focus: Number and Operations - Fractions Grade Four: Multiply a fraction by a whole number Grade Five:Multiply a fraction by a fraction Divide unit fractions by a whole number; and whole numbers by unit fractions Grade Six:Interpret and compute quotients of fractions and solve word problems
Conceptual and Procedural Understanding Describe how you would solve this problem?
Shift 2: Coherence Coherence provides the opportunity for students to make connections between mathematical ideas and across content areas –Each standard is not a new event, but an extension of previous learning –Occurs both within a grade and across grades –Allows students to see mathematics as inter-connected ideas Mathematics instruction cannot be relegated to merely a checklist of topics to cover, but instead must be centered around a set of interrelated and powerful ideas
Take the number apart? Tina, Emma, and Jen discuss this expression: Tina: I know a way to multiply with a mixed number, like that is different from the one we learned in class. I call my way “take the number apart.” I’ll show you.
Which of the three girls do you think is right? Justify your answer mathematically. First, I multiply the 5 by the 6 and get 30. Then I multiply the by the 6 and get 2. Finally, I add the 30 and the 2, which is 32. –Tina: It works whenever I have to multiply a mixed number by a whole number. –Emma: Sorry Tina, but that answer is wrong! –Jen: No, Tina’s answer is right for this one problem, but “take the number apart” doesn’t work for other fraction problems.
Example explanation Why does 5 x 6 = (6x5) + (6 x ) ? Because 5 1/3 = 5 + 1/3 6(5 1/3) = 6(5 + 1/3) = (6x5) + (6x1/3) because a(b + c) = ab + ac
Coherence At the high school level, students relate their previous understandings as they learn to multiply binomials (3x + 5)(2x + 6) 3x(2x + 6) + 5 (2x + 6)
Mathematical Practices Connected directly to the content (happens in the context of solving real problems)
William Schmidt Keynote speaker, July 2012, Curriculum/Research Director’s Meeting –CCSS in Mathematics Can potentially elevate the academic performance of our students Standards relationship to student achievement is influenced by the instructional materials/units available for teachers to use
Implementation Good News 90% of teachers are positive about the CCSS Bad News 80% of teachers indicated that the CCSSM is pretty much the same as their state standards. They indicated that they would keep teaching a topic in their grade level even if not in the Standards
Ugly news Teachers self-report Grades 1-5: only ½ felt well-prepared to teach the standards; Grades 6-8; only 60% feel well-prepared; HS; only 70% feel well prepared
What criteria will you use to review and select materials/resources Now what?
Reviewing secondary materials The degree to which specific trajectories of mathematics topics are incorporated appropriately across grade-band curriculum materials. The curriculum materials support the development of students’ mathematical understanding
Reviewing secondary materials The curriculum materials support the development of students’ proficiency with procedural skills. The curriculum materials assist students in building connections between mathematical understanding and procedural skills.
Reviewing secondary materials Student activities build on each other within and across grades in a logical way that supports mathematical understanding and procedural skills. The materials provide opportunities for students to develop the Standards for Mathematical Practice as “habits of mind” (ways of thinking about mathematics that are rich, challenging, and useful) throughout the development of the Content Standards.
A lot to do…. Materials/resources/professional development