Key Shifts in Mathematics

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: Focus:Greaterfocusonfewertopics.

The Common Core calls for greater focus in mathematics. Rather than racing to cover many topics in a mile-wide, inch-deep 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:

Grades K-2: Concepts, skills, and problem solving related to addition and subtraction Grades 3-5: Concepts, skills, and problem solving related to multiplication and division of whole numbers and fractions Grades 6: Ratios and proportional relationships, and early algebraic expressions and equations Grades 7: Ratios and proportional relationships, and arithmetic of rational numbers Grades 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:Linkingtopicsandthinkingacrossgrades.

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 4th 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 5th 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 grade-level word problems in which students apply mathematical skills to solve problems. Rigor:Pursueconceptualunderstanding,proceduralskillsandfluency,andapplicationwithequalintensity.

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 single-digit 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. Source: http://www.corestandards.org/other-resources/key-shifts-in-mathematics

Operations and Algebraic Thinking

The Hunt Institute

Jason Zimba, Lead Writers, Common Core State Standards for Mathematics The Common Core Standards focus on Number and Operations in early grades. And number and operations is usually a single strand in state standards, but in the Common Core Standards there are three domains in this area: Operations and Algebraic Thinking, Number and Operations in Base Ten, and Number and Operations with Fractions. One reason to distinguish these three things is to highlight the sense in which operations are the same no matter what students are operating with. Addition, subtraction, multiplication, and division have meanings, mathematical properties, and uses that transcend the particular sorts of objects that one is operating on, whether those be multi-digit numbers or fractions or variables or variable expressions. And so operations and algebraic thinking domain is the one that evolves most directly into middle grades expressions and equations. The strategies and algorithms that students use to add multi-digit numbers differ from the strategies and algorithms that they use to add fractions, but in both cases they are adding; they are using an operation of joining or increasing. They are using an operation whose meanings and uses and mathematical properties are the same in either case. And in the domain of Operations and Algebraic Thinking, it is those meanings, properties, and uses which are the focus. And it is those meanings, properties, and uses that will remain when students begin doing algebra in middle grades. So arithmetic in the Common Core Standards is both an important skill set, as well as a rehearsal for algebra. Source: https://www.youtube.com/watch?v=ONPADo_Nt14&list=PLD7F4C7DE7CB3D2E6 http://www.ccsso.org/Resources/Digital_Resources/Common_Core_Implementation_Video_Series.html https://sites.google.com/site/commoncoreinvermont/home/hunt-institute-video/hunt-videos-math

Gathering Momentum for Algebra

The Hunt Institute

William McCallum, Lead Writers, Common Core State Standards for Mathematics 0:02 0:15 They go along through number and operations, they learn geometry, and then 0:20 when they get to algebra, it is if they have to climb this cliff. 0:23 We tried to write the standards so that there's a ramp up to the top of that cliff. 0:27 We started all the way back in kindergarten 0:31 thinking about what it is that kids are doing with number 0:34 that prepares them for algebra. And that's why we made this choice to separate out 0:38 what is traditionally just being a number and operation strand 0:42 into two domains. Operations and algebraic thinking, 0:45 which is where kids learn about how operations work. 0:49 For example, that's where they learn that subtraction is really the solution to a 0:54 missing addend problem, so there's that relationship between addition and subtraction. 1:00 Later on they learn that there's a similar relationship between multiplication and division. 1:04 That domain is the one that's leading them up to expressions and equations in 1:09 middle school. At the same time, there is this number and operations in base ten 1:13 domain which is where they are learning to compute 1:14 and do addition and subtraction. As distinct from 1:18 understanding the operation itself, they are learning how to do the calculation 1:22 That's important to build number sense and to build facility [fluency]. 1:25 That's the one that leads into this number system domain in middle school. 1:28 Fractions coming in a very important way in third grade. 1:32 Fractions are another aspect of this ramp up to algebra, because 1:36 if you think about it when kids start working with fractions they are almost doing 1:39 algebra with numbers, really. 1:41 They see a fraction 3 over 4 and they have to add it to four-fifths. 1:45 It's almost as if they're adding A over B to C over D. 1:49 They see those operations starting to play out with how they work with numbers as a 1:53 rehearsal for doing it later when they get to algebra. 2:05 Source: https://www.youtube.com/watch?v=ONPADo_Nt14&list=PLD7F4C7DE7CB3D2E6 http://www.ccsso.org/Resources/Digital_Resources/Common_Core_Implementation_Video_Series.html https://sites.google.com/site/commoncoreinvermont/home/hunt-institute-video/hunt-videos-math

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