Strategies for Common Core Mathematics

Implementing the Standards for Mathematical Practice, 9-12

Leslie Texas, Tammy Jones

• Publisher: Routledge
• Year: 2013
• eBook ISBN: 9781317921349
• Edition: 1
• Topic: Education
• Subtopic: Education General

This new, practical book provides an explanation of each of the eight mathematical practices and gives high school educators specific instructional strategies that align with the Common Core State Standards for Mathematics.

Math teachers, curriculum coordinators, and district math supervisors get practical ideas on how to engage high school students in mathematical practices, develop problem-solving skills, and promote higher-order thinking. Learn how to scaffold activities across grades and get strategies you can implement immediately in your classroom. All high school mathematics educators should have this book in their professional libraries!

Section 1

The Doorway to the Common Core

The Common Core State Standards for Mathematics

With the creation of the Common Core State Standards for Mathematics (CCSSM), a new era in mathematics education began for those states that adopted them. States that have adopted the CCSSM now have a common goal in mathematics education.

Building on the excellent foundation of standards states have laid, the Common Core State Standards are the first step in providing our young people with a high-quality education. It should be clear to every student, parent, and teacher what the standards of success are in every school. (Common Core State Standards Initiative, 2012a)

How clear is it “what the standards of success are”? The standards were written “to provide a clear and consistent framework to prepare our children for college and the workforce” (Common Core State Standards Initiative, 2012a). In its Myths vs Facts section, the CORE Standards website (www.corestandards.org/about-the-standards/myths-vs-facts) states that the standards “are not a curriculum. They are a clear set of shared goals and expectations for what knowledge and skills will help our students succeed.” What are the implications for the classroom teacher, whether teaching kindergarten or high-school algebra? Teachers need to become fluent in the content not only to teach their grade and course, but also to reinforce the prior content knowledge of their students and understand how the current content supports where the students are going. Only through studying these progressions will teachers truly be able to connect the mathematics they are teaching to what their students have previously learned and to what will be expected of them in upcoming grades.

The Standards for Mathematical Practice

According to the Common Core State Standards for Mathematics, “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students” (2012b). The National Council of Teachers of Mathematics (NCTM), in its Principles and Standards for School Mathematics (PSSM), states that “the five Process Standards highlight ways of acquiring and applying content knowledge” (2005, p. 29).

The Standards for Mathematical Practice (SMP) are based upon the NCTM process standards and the strands of mathematical proficiency in the National Research Council’s report Adding It Up. NCTM chose to present its mathematical processes from the point of view that these are a collection of best practices that teachers can utilize to help their students develop a depth of understanding of key mathematical concepts that also leads to increased retention of those concepts.

Here are the eight Standards for Mathematical Practice (Common Core State Standards Initiative, 2012b):

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

The content standards provide the context, whereas the Standards for Mathematical Practice assist students in developing mathematical proficiency. The eight practices are distinct from one another but interconnected in ways that support students in becoming mathematically proficient. Expertise is generated in practice but implemented through process.

Expertise is generated in practice but implemented through process.

These practices need to be incorporated into daily classroom instruction. The creators of the Common Core State Standards for Mathematics took the perspective that the Standards for Mathematical Practice are observable indicators of student understanding that identify the level of expertise that teachers should foster in their students. The writers for the CCSSM even argue that a lack of understanding in the mathematical content inhibits students from participating in the mathematical practices. But it is the practices themselves that help develop that understanding. So connecting the content to the mathematical practices is critical if teachers want to develop solid mathematical proficiency in their students (Common Core State Standards Initiative, 2012b).

Will these mathematical practices look the same in a kindergarten classroom as in a high-school mathematics classroom? Not necessarily. The students are at a different level in their journey toward attaining expertise in various mathematical topics and skills. Students need procedural fluency in a topic as well as an understanding of the concept. It is for this reason there are three books in this series so these differences can be addressed specifically for each grade band.

The processes are how a student gains proficiencies in the content that allows them to develop the practices that ultimately carry them through their mathematical journey. The CCSSM are a collection of processes, proficiencies, and practices that produces students who are ready for successful transition into the workplace or college.

The Standards for Mathematical Practice versus the Content Standards

The Standards for Mathematical Practice identify the habits that mathematically proficient students have developed. Habits are developed over time. How long it takes to develop a habit is debatable. But clearly, mathematically proficient students, those who have developed these eight habits, have experienced mathematics regularly and consistently over a period of time.

The Standards for Mathematical Practice are how the student engages with the mathematical content to develop both procedural fluency and conceptual understanding. They are separate, yet must be developed together to ensure that students can effectively understand the content and engage in the practices. The processes are how students gain proficiencies in the content that allow them to develop the practices that ultimately become sound habits.

The processes are how students gain proficiencies in the content that allow them to develop the practices that ultimately become sound habits.

How the Standards for Mathematical Practice Support the Content Standards

For students to connect the practices to the content, teachers need to understand how students learn mathematics and that not all students learn the same way or in the same time frame. Teachers will need to provide opportunities for students to delve deeply into a concept by designing lessons that explicitly embed and utilize the Standards for Mathematical Practice.

The eight Standards for Mathematical Practice are not experienced in isolation. In fact, most of the time, students simultaneously employ several of the practices as they engage in mathematical experiences. If students are to “construct viable arguments and critique the reasoning of others,” they will need to “attend to precision” by using precise vocabulary and symbolism. They will then check the reasonableness of their solutions by gathering supporting evidence. Students who “look for and make use of structure” will also “look for and express regularity in repeated reasoning” while they “make sense of problems and persevere in solving them.” Along the way they also “use appropriate tools strategically.”

The content standards also support the practices. The writers identify “potential ‘points of intersection’” between the content and the SMP as places where the content mastery requires a level of deeper understanding.

The Partnership for Assessment of Readiness for College and Careers (PARCC) Content Framework for Mathematics notes that “opportunities for in-depth work on key concepts and connections to critical practices … intend to support… efforts to deliver instruction that connects content and practices while achieving the standards’ balance of conceptual understanding, procedural skill and fluency, and application” (n.d., para. 3).

Teaching the Standards for Mathematical Practice

Teaching students to become mathematical thinkers does not happen randomly. In order for students to meet the expectations of high-level content knowledge contained in the CCSSM, it is necessary for the students to build a foundation of thinking and communicating mathematically. These practices, outlined in the Standards for Mathematical Practice, must be explicitly and intentionally designed into the curriculum and become a focus of instructional practice in the classroom.

The next section contains strategies for teaching the mathematical practices while simultaneously addressing the content of the Common Core State Standards for Mathematics. These ideas were chosen for their flexibility: they can be taught at any grade level and address almost any concept. In addition, they are easy to prepare and implement. This allows for continued use and refinement while, at the same time, not requiring an added burden of hours of preparation. The section contains an overall description as well as detailed directions for implementation of each strategy. See the Strategies Matrix (page 8) for an overview of mapping each strategy to the SMP.

The Doorway to the Standards for Mathematical Practice

The Standards for Mathematical Practice can be seen as the doorway to implementing the Common Core State Standards for Mathematics. Students, as well as educators and administrators, need to understand what these eight practices entail and what they might look like in their classrooms and mathematical experiences. These standards can be grouped into various clusters to represent differing foci. For the purpose of developing strategies to support the implementation of the CCSSM, the SMP can be grouped as shown in the doorway graphic (page 6).

A door provides the first impression of what lies beyond. When the door is open, it invites one to enter and experience what is behind it. When a door is closed, it evokes a sense of mystery and the unknown. For some educators, the SMP are an open door to the CCSSM. These educators are familiar with implementing...

Table of contents

• Cover
• Title Page
• Copyright Page
• Acknowledgments
• Table of Contents
• Foreword
• Preface: A Note to Our Readers
• Section 1: The Doorway to the Common Core
• Section 2: Framing Strategies for Implementing SMPs #1, #5, and #6
• Section 3: Strategies for Implementing Combinations of SMPs
• Appendix A: Blackline Masters for Strategies and Illustrations
• Appendix B: Common Core State Standards for Mathematics Resources
• References