Mathematics Progressions – Common Core

1. Mathematics Progressions – Common Core Elizabeth Gamino, Fresno Unified Public Schools Denise Walston, Council of the Great City Schools

2. Purpose • Review and take a closer look at the Mathematics Progressions • Focus, Coherence, and Rigor • Instructional implications • Scaffolding • Explain their thinking; critique the reasoning of others • Developing mathematical arguments • Reading and interpreting real-world problems • Technical language of the discipline

3. Instructional Shifts Mathematics A Look Back

4. Mathematics Instructional Shifts • Focus • Coherence • Rigor • Conceptual understanding • Procedural skill and fluency • Applications

5. Focus • Significantly narrow and deepen the way that time and energy is spent in the math classroom • Communicate focus so that it is manageable in instruction; it is more than merely writing a standard a day • Focus deeply on those concepts emphasized in the standards • Provide the time for students to transfer mathematical skills and understanding across concepts and grade levels • Deep conceptual understanding • Connect conceptual and procedural understanding • Transitions from concrete↔pictorial↔language↔abstract

6. Coherence • Coherence provides the opportunity for students to make connections between mathematical ideas and across content areas • Connects the learning both within a grade and across grades • Thinking across grades • Each standard is not a new event, but an extension of previous learning • 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, rather than a series of disconnected topics

7. Rigor With equal intensity • Conceptual Understanding • Involves more than getting the right answer • Access concepts from multiple perspectives • Transitions from concrete↔pictorial↔language↔abstract • Procedural Skill and Fluency • Study algorithms as a way to see the structure of mathematics (organization, patterns, predictability) or apply a variety of appropriate procedure flexibly to solve problems • Students are expected to achieve speed and accuracy with simple calculations (at specific grade levels) • Fluent is used in the Standards to mean “efficient and accurate” • Class time and/or homework should be structured for students to practice core functions such as single-digit multiplications • Application • Expectation that students apply math and choose the appropriate concept for application, even when not prompted to do so • Apply math concepts in real-world situations • Mathematical modeling

8. Common Core State Standards Mathematics Closer Look at Fractions

9. Fraction Concepts • Grade One: Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrase half of, fourth of, and quarter of. Describe the whole as two of or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

10. Fraction Concepts • Grade Two Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

11. Coherence – 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)

12. Coherence: 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

13. The Mathematics Progressions Developing a deep understanding of focus, coherence, and rigor

14. Mathematics Progressions • Common Core State Standards in Mathematics • Informed by research on children’s cognitive development and the logical structure of mathematics • Narrative documents • Progression of concepts across several grade bands

15. Closer Look – Fraction Progression

16. Fraction as number

17. Linking content to the Mathematical Practices

18. and District Perspective • Read and analyze the progressions (book study, close read) • Through the lens of focus, coherenceandrigor • Need to look at the standards in their entirety K-12 • Determine what happens prior to students entering your door and where they need to go - beyond the grade level below and above.

19. Problems for consideration Exploring the Fraction Progression using illustrative tasks

21. Questions to consider • What are some things that you had to attend to? • What are some things that ELLs need to be aware of? • What are the implications for planning for instruction? • How might you assess student understanding specifically related to ELLs? • What are the implications for professional development?

24. Online Resources Fraction Professional Development Module

25. Fraction Progression Online Module • Goals/Purpose: • Deepen educators content knowledge, specifically around the mathematics “Fractions” progression; • Engage practicing educators in the development of CCSS-aligned professional development; and • Provide consistent, high-quality professional development that can be used at large scale and online across PARCC states to inform elementary teachers and instructional leaders • Partnership with CGCS, Institute for Mathematics & Education, University of Arizona, and Achieve

26. Design within Edmodo • Features of the module • Online and interactive • Based on a framework developed collaboratively with CGCS • Illustrative of the Fractions Progression in the CCSSM • Seven units anchored by a 3-5 minute video • Three to four illustrative tasks associated with the progression • Built-in, interactive checks for understanding • Interactive-check for understandings – Quiz – linked to the commentary

27. Online Professional Development Module • http://commoncoreworks.org/domain/121