Key Shifts in Mathematics
Introduction
The Common Core State Standards for Mathematics build on the best of existing standards and reflect the skills and knowledge students will need to succeed in college, career, and life. Understanding how the standards differ from previous standards—and the necessary shifts they call for—is essential to implementing them.
The following are the key shifts called for by the Common Core:

Greater focus on fewer topics
The Common Core calls for greater focus in mathematics. Rather than racing to cover many topics in a milewide, inchdeep curriculum, the standards ask math teachers to significantly narrow and deepen the way time and energy are spent in the classroom. This means focusing deeply on the major work of each grade as follows:
 In grades K–2: Concepts, skills, and problem solving related to addition and subtraction
 In grades 3–5: Concepts, skills, and problem solving related to multiplication and division of whole numbers and fractions
 In grade 6: Ratios and proportional relationships, and early algebraic expressions and equations
 In grade 7: Ratios and proportional relationships, and arithmetic of rational numbers
 In grade 8: Linear algebra and linear functions
This focus will help students gain strong foundations, including a solid understanding of concepts, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the classroom.

Coherence: Linking topics and thinking across grades
Mathematics is not a list of disconnected topics, tricks, or mnemonics; it is a coherent body of knowledge made up of interconnected concepts. Therefore, the standards are designed around coherent progressions from grade to grade. Learning is carefully connected across grades so that students can build new understanding onto foundations built in previous years. For example, in 4^{th} grade, students must “apply and extend previous understandings of multiplication to multiply a fraction by a whole number” (Standard 4.NF.4). This extends to 5^{th} grade, when students are expected to build on that skill to “apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction” (Standard 5.NF.4). Each standard is not a new event, but an extension of previous learning.
Coherence is also built into the standards in how they reinforce a major topic in a grade by utilizing supporting, complementary topics. For example, instead of presenting the topic of data displays as an end in itself, the topic is used to support gradelevel word problems in which students apply mathematical skills to solve problems.

Rigor: Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity
Rigor refers to deep, authentic command of mathematical concepts, not making math harder or introducing topics at earlier grades. To help students meet the standards, educators will need to pursue, with equal intensity, three aspects of rigor in the major work of each grade: conceptual understanding, procedural skills and fluency, and application.
Conceptual understanding: The standards call for conceptual understanding of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives in order to see math as more than a set of mnemonics or discrete procedures.
Procedural skills and fluency: The standards call for speed and accuracy in calculation. Students must practice core functions, such as singledigit multiplication, in order to have access to more complex concepts and procedures. Fluency must be addressed in the classroom or through supporting materials, as some students might require more practice than others.
Application: The standards call for students to use math in situations that require mathematical knowledge. Correctly applying mathematical knowledge depends on students having a solid conceptual understanding and procedural fluency.