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by Amitra Schwols, Kathleen Dempsey and John Kendall
Table of Contents
The Common Core mathematics standards are organized into two sets: the Standards for Mathematical Content, designed to cross traditional course boundaries and cover all the conceptual mathematical understandings students need to develop from kindergarten through 12th grade, and the Standards for Mathematical Practice, which highlight the kinds of expertise that students must develop and use throughout this same grade span.
As we will show in this guide, the Common Core standards differ in many ways from most existing state standards documents, providing a greater level of detail about concepts, thought processes, and approaches. This level of detail often leads to much longer, more involved standards, some of which are up to a paragraph in length. Some of the standards detail conceptual methods of teaching and learning skills and concepts (e.g., applying the properties of operations to generate equivalent expressions, understanding that solving an equation or inequality is a process of answering a question). This is in stark contrast to many prior sets of state standards, which were far less explicit and typically used a single sentence to describe the skills and knowledge required of students. Another example of this detailed focus on the mental processes required in understanding mathematical concepts is found in the set of Standards for Mathematical Practice, which receives the same level of emphasis as the Standards for Mathematical Content.
In this chapter, we will walk you through the standards' structure, provide an overview of how the middle school mathematics standards fit together, and offer some guidance on what to focus on as you begin your implementation efforts.
At the middle school level, the Standards for Mathematical Content are organized first by grade, then by domain, and finally by cluster. Each grade level's set of standards is introduced with a one- or two-page introduction, which consists of two parts—a summary of the three to four critical areas (topics) for each grade, and an in-depth narrative description of those critical areas. Figure 1.1 provides a brief, grade-by-grade summary of the critical areas for middle school.
Figure 1.1. Critical Areas Within the Middle School Mathematics Domains by Grade Level
Domain Name
Grade 6
Grade 7
Grade 8
Ratios and Proportional Relationships
Connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems
Developing understanding of and applying proportional relationships
Domain not addressed at this grade level
The Number System
Completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers
Developing understanding of operations with rational numbers and working with expressions and linear equations
No critical areas identified for this domain at this grade level
Expressions and Equations
Writing, interpreting, and using expressions and equations
Formulating and reasoning about expressions and equations, including solving linear equations and systems of linear equations
Functions
Domain not addressed at these grade levels
Grasping the concept of a function and using functions to describe quantitative relationships
Geometry
Solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume
Analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem
Statistics and Probability
Developing understanding of statistical thinking
Drawing inferences about populations based on samples
Note: Content in this table was adapted from the descriptions in the standards' grade-level introductions.
One important aspect of the introductions is that their narrative descriptions of the critical areas also provide insight into the meanings or limitations of the standards that are not immediately apparent to someone just reading the standards alone. For example, the first standard in the 6th grade Ratios and Proportions domain asks students to "understand the concept of a ratio"—quite a broad aim. The introduction to the 6th grade standards (CCSSI, 2010c, p. 39) provides vital clarification, articulating in greater detail which aspects of the concepts of ratio a 6th grade student needs to understand, connecting ratios, rates, and an ability to reason with multiplication and division. Although it can be tempting to skim introductory text, teachers and administrators should take the time to review the standards' grade-level introductions thoroughly to ensure that they get all the information about the standards that is available.
After each grade-level introduction, the standards are organized hierarchically, as follows:
This guide contains one chapter for each of the six middle school domains, and at the beginning of each chapter, you will find a chart that provides an overview of each grade level's clusters and standards. In those charts, and throughout this guide, we will be referencing the content standards using a slightly abbreviated version of the CCSSI's official identification system, which provides a unique identifier for each standard in the Common Core and can be very useful for school staffs when developing crosswalks, planning lessons, and sharing lesson plans. Under this system, all mathematics content standards begin with the formal prefix "CCSS.Math.Content"; we have dropped this prefix in our references throughout the guide, including the sample lessons. The next piece of the code for standards in grades K–8 is the specific grade level, which is followed by the domain abbreviation, the letter identifying the particular cluster within the domain, and then the specific standard number. For example, "8.G.B.6" is shorthand for Grade 8, Geometry (the domain name), Cluster B (of the domain's three clusters, identified A–C), Standard 6. "8.G.B" is a shorthand way of referring to all the standards within Cluster B of the 8th grade Geometry domain. Note that codes for high school mathematics standards follow the "CCSS.Math.Content" prefix with the letters "HS" and the abbreviation for the conceptual category before continuing with domain, cluster, and standard identifiers. For example, "HSA-REI.A.1" refers to Standard 1 within the first cluster ("Understand solving equations as a process of reasoning and explain the reasoning") of the Reasoning with Equations and Inequalities domain of the high school algebra standards.
Taken as a whole, the Common Core's mathematical content standards at the middle school level identify what students should know and be able to do in order to be prepared for mathematics study at the high school level and, ultimately, to be college and career ready.
Emphasis on students' conceptual understanding of mathematics is an aspect of the Common Core standards that sets them apart from many state standards. However, the eight Standards for Mathematical Practice, listed in Figure 1.2, play an important role in ensuring that students are engaged in the actual use of mathematics, not just in the acquisition of knowledge about the discipline. Indeed, the table of contents in the standards document gives equal weight to the Standards for Mathematical Practice and to the Standards for Mathematical Content. This dual focus, echoed throughout the standards document's introductory material, has been undertaken to ensure the standards "describe varieties of expertise that mathematics educators at all levels should seek to develop in their students" (CCSSI, 2010c, p. 6).
Figure 1.2. The Standards for Mathematical Practice
MP1. Make sense of problems and persevere in solving them.
MP2. Reason abstractly and quantitatively.
MP3. Construct viable arguments and critique the reasoning of others.
MP4. Model with mathematics.
MP5. Use appropriate tools strategically.
MP6. Attend to precision.
MP7. Look for and make use of structure.
MP8. Look for and express regularity in repeated reasoning.
The writers of the Common Core describe these practice standards in an introduction, explaining that the standards are derived from the process standards of the National Council of Teachers of Mathematics (NCTM, 2000)and the strands of mathematical proficiency found in the National Research Council report Adding It Up (2001). A brief description of the meaning of the practice standards is provided in the front of the mathematical standards document (CCSSI, 2010c, pp. 6–8).
In addition to stressing mathematics proficiencies that cross all domains, the mathematical practice standards ensure that students who are focused on skills and processes don't find themselves engaged in rote activities that provide them no deeper sense of how mathematics works as an integrated whole. For example, solving simple equations or inequalities might be seen as nothing more than a series of steps. In the past, standards documents required nothing more than that the process be taught, which meant that often students were shown processes (such as the substitution method) and expected to memorize them. What students were not expressly expected to do was gain a deep understanding of the reasoning and meaning behind the processes. In contrast, the Common Core standards require that students understand that solving equations and inequalities means answering the question "Which values from a specified set make the equation or inequality true?" The Common Core standards ask that students possess this understanding in addition to being able to use substitution (6.EE.B.5). The underlying rationale is that students who are able to articulate this understanding are positioned to gain a deeper understanding of equations and inequalities, which allows them to see the utility of the process over a wider range of problems.
Please note that, as with the content standards, the mathematical practice standards have official identifiers, which we have shortened in this guide's sample lessons. For example, we abbreviate Mathematical Practice 1, officially "CCSS.Math.Practice.MP1," as "MP1."
A recent survey of more than 13,000 K–12 math teachers and 600 district curriculum directors across 40 states shows that teachers are highly supportive of the Common Core standards. That's the good news. On the other hand, the same survey shows that an overwhelming majority (80%) mistakenly believe that the standards are "pretty much the same" as their former state standards, and only about 25 percent of respondents are willing to stop teaching a topic that they currently teach, even if the Common Core State Standards do not support teaching that topic in their current grade (Schmidt, 2012).
These findings suggest some damaging possible consequences. If teachers don't recognize the Common Core's new emphasis on depth of content understanding, they may not take the steps necessary to narrow the focus of their curriculum so that students will have the time they need to develop that deeper understanding. Furthermore, teachers' unwillingness to stop teaching familiar or favorite content that the standards do not require reinforces the possibility that, while students may be exposed to a wide variety of mathematical concepts, they will not reach the required level of mastery set for concepts that have been identified as critical.
We want to highlight two documents that can provide significant support for teachers' instructional efforts. The first is Appendix A of the standards document (CCSSI, 2010d), which includes information on an accelerated pathway for middle school. This pathway illustrates how to compress the contents of a first-year high school mathematics course (Algebra I or Integrated Pathway: Mathematics I) into middle school. The second useful document is Progressions for the Common Core State Standards in Mathematics (Common Core Standards Writing Team, 2011). Still in draft form at the time of this writing, it details some useful strategies for teaching the middle school mathematics standards, and we urge anyone interested in specific strategies and examples to read it.
As noted, the standards document and its appendix do offer some ideas for how to get started planning instruction and teaching the standards, and here in this guide, we share our own best advice.
The Standards for Mathematical Practice are one of the potentially challenging aspects of Common Core implementation. As described on page 6, the mathematical practice standards are found in two places in the standards document: in the document's introduction and in the overview of each grade. The guidance found in the main introduction provides valuable insight into each mathematical practice standard, and we recommend that teachers become extremely familiar with these descriptions and spend some time planning how to incorporate the practices into each course. In the chapters to come, we offer our own ideas about how teachers might integrate the mathematical practice standards with each of the domains in the mathematical content standards.
By sharpening the focus of each grade on three to four critical areas identified by the Common Core writers (as described on pp. 7–8), teachers can develop students' understanding of those concepts to a degree that's deeper than previous standards required or allowed. The outcome is stronger foundational knowledge.
Remember that the Common Core mathematics standards are designed to be coherent within and across grades. The chapters on the domains clarify how the concepts found in the middle school standards are organized across grades, underscoring that each standard is best understood not as new knowledge but as extensions of ideas presented in previous school years.
The Standards for Mathematical Practice provide further connective tissue between the standards at each grade level and within the various domains, which we highlight throughout this guide. However, it is important to stress that what we present are only a few examples of such connections; we do not mean to suggest that no other connections can or should be made. We encourage teachers to build on the proposals here to strengthen their own practice and enhance their implementation of the Common Core standards.
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Now that we've looked at the overall structure of the Common Core standards for middle school mathematics, we will examine each domain, addressing the specific standards at the various grade levels.
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