K–8 Session 3: Exploring the Domain Progressions in Grades K–8 Module 1: A Closer Look at the Common Core State Standards for Mathematics

Expected Outcomes Build understanding of the mathematical concepts within each domain and how they progress across grades. Build understanding of the coherence in the standards. Discuss how the progressions in the standards can be used to inform teaching and learning. 2

Principle #1: Increases in student learning occur only as a consequence of improvements in the level of content, teachers’ knowledge and skill, and student engagement. Richard Elmore, Ph.D., Harvard Graduate School of Education Principle #2: If you change one element of the instructional core, you have to change the other two. The Instructional Core 3

Adapted from the Public Education Leadership Project at Harvard University STRUCTURES POLICIES, PROCESSES & PROCEDURES RESOURCES HUMAN, MATERIAL, MONEY STAKEHOLDERS CULTURE Organizational Elements 4

Focused and Coherent For over a decade, research studies of mathematics education in high-performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on the promise of common standards, the standards must address the problem of a curriculum that is ‘a mile wide and an inch deep.’ - Common Core State Standards for Mathematics, page 3 5

Coherence Content standards and curricula are coherent if they are … articulated over time as a sequence of topics and performances that are logical and reflect … the sequential or hierarchical nature of the disciplinary content … What and how students are taught should reflect not only the topics that fall within a certain academic discipline, but also the key ideas that determine how knowledge is organized and generated within that discipline. - Common Core State Standards for Mathematics, page 3 6

These Standards endeavor to follow such a design, not only by stressing conceptual understanding of key ideas, but also by continually returning to organizing principles such as place value or the laws of arithmetic to structure those ideas. - Common Core State Standards for Mathematics, page 4 7

For each grade level from kindergarten through grade 8, the Critical Areas outline the essential mathematical ideas for each grade level. 8

Domain 9

Cluster Headings Standards 10

Domains for K-8 11 K Counting and Cardinality Operations and Algebraic Thinking Number and Operations in Base Ten Number and Operations - Fractions Ratios and Proportional Relationships The Number System Expressions and Equations Functions Geometry Measurement and Data Statistics and Probability

Domains for K-8 12

Exploring the Domains Review the clusters of standards within the domain. Start with the lowest grade for which the domain begins and cut out each cluster of standards. Start with one cluster and explore how the concept progresses through the grades. How does the concept change and increase in rigor and complexity for the student? Identify new concepts that are introduced in subsequent grades and follow them through the years. Paste your work on a large sheet of chart paper. Use different colored arrows and lines to show, and words to describe, how each concept or standard progresses from one year to the next. 13

Small Group Discussion Where are each of your students in the progression? What supports can you use to accelerate learning of students in an effort to bring their understandings and skills to the appropriate level or to go deeper into the content? What connections can you find between this domain and the other domains? 14

Whole Group Discussion What are your general impressions of how the content progresses from one grade to the next? To what extent do the standards in the domain address the concept of coherence? What are some common student misconceptions within this domain? How would you use the ideas you explored in this session to inform student assessment and your instruction? 15

… the ‘sequence of topics and performances’ that is outlined in a body of mathematics standards must also respect what is known about how students learn. … developing ‘sequenced obstacles and challenges for students … absent the insights about meaning that derive from careful study of learning, would be unfortunate and unwise.’ In recognition of this, the development of these Standards began with research- based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time. - Common Core State Standards for Mathematics, page 4 16

One promise of common state standards is that over time they will allow research on learning progressions to inform and improve the design of standards to a much greater extent than is possible today. - Common Core State Standards for Mathematics, page 5 17

Expected Outcomes Build understanding of the mathematical concepts within each domain and how they progress across grades. Build understanding of the coherence in the standards. Discuss how the progressions in the standards can be used to inform teaching and learning. 18

Reflection Revisit the meaning of “coherence.” To what extent does the domain you explored bring coherence to the standards? How will your knowledge of the standards progression in this domain inform your curriculum and guide your instruction? What questions do you still have about this domain? How has this activity increased your understanding of the instructional core? 19