A Teacher's Guide to Using the Common Core State Standards With Mathematically Gifted and Advanced Learners

National Assoc For Gifted Children, Gail R. Ryser, Susan Assouline

• Publisher: Routledge
• Year: 2021
• eBook ISBN: 9781000491548
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

A Teacher's Guide to Using the Common Core State Standards in Mathematics provides teachers and administrators with practical examples of ways to build a comprehensive, coherent, and continuous set of learning experiences for gifted and advanced students. It describes informal, traditional, off-level, and 21st century math assessments that are useful in making educational decisions about placement and programming. Featuring learning experiences for each grade within one math progression, the book offers insight into useful ways of both accelerating and enriching the CCSS mathematics standards. Each of the learning experiences includes a sequence of activities, implementation examples, and formative assessments. Specific instructional and management strategies for implementing the standards within the classroom, school, and school district will be helpful for both K-12 teachers and administrators.

Chapter 1

Overview

DOI: 10.4324/9781003232636-1

In this chapter, we provide an overview of this book and how the Common Core State Standards might be implemented with learners who are gifted and advanced in mathematics. Specifically, we address standards, differentiation, scope and sequence, assessments, differentiated learning experiences, and management strategies within the classroom, school, and district.

Standards

Within the past 30 years, standards have assumed an important role in schools. They define the important knowledge and skills within a specific subject area and provide guidelines for the development of curriculum, assessments, and critical benchmarks for students. Understanding the domain along with evidence-based instructional practices can help individual teachers establish a set of clear expectations and ultimately improve student outcomes. Common standards also provide consistency so that “no matter where [students] live, [they] are well-prepared with the skills and knowledge necessary to collaborate and compete with their peers in the United States and abroad” (NGA & CCSSO, 2010c). In this section, we will discuss the Common Core State Standards for Mathematics and their alignment to 21st century skills (Partnership for 21st Century Skills, n.d.) and the NAGC Gifted Education Programming Standards (NAGC, 2010).

Common Core State Standards for Mathematics

Adopted by 45 states, the District of Columbia, and four territories to this date, the Common Core State Standards for Mathematics (CCSSM; NGA & CCSSO, 2010a, 2010b) have quickly become the foundation for developing learning activities in mathematics. Designed by teams of math specialists across states, the new standards are intended to prepare K–12 students for college and the workplace. They emphasize thinking, problem solving, collaboration, and communication and are informed by research and reports from national and international studies such as the National Assessment of Educational Progress (NAEP, 2011) Mathematics Framework and the Trends in International Mathematics and Science Study (TIMSS) report in mathematics (National Center for Education Statistics [NCES], 2007).

In mathematics, two sets of standards are described: Standards for Mathematical Content (CCSSM-C) and Standards for Mathematical Practice (CCSSM-P). The CCSSM-C are organized by grade and secondary levels, standards, clusters, and domains. Standards define what students should understand and be able to do, clusters summarize groups of related standards, and domains are larger groups of related standards. For example, at the fifth-grade level, within the domain of Operations and Algebraic Thinking (5.OA), the student is expected to “write and interpret numerical expressions” (cluster heading) by “using parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols” and by “writing simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them” (standards; NGA & CCSSO, 2010a, p. 35).

It is important to notice that the standards and clusters may be related to other domains at the same grade level or across domains at different levels, forming a learning progression. For example, in measurement, students use measurable attributes to describe and compare objects, situations, or events at the elementary level; use measurable attributes in models and formulas at the middle school level; and explore measurement systems and measurement of more complex or abstract quantities at the high school level. Similarly, graphing is used to represent data in the Measurement and Data domain, to solve problems in the Geometry domain by graphing points on the coordinate plane, and to analyze patterns and relationships in the Operations and Algebraic Thinking domain by graphing ordered pairs on a coordinate plane.

Therefore, when educators are teaching the Common Core, they need to be cognizant of vertical and lateral alignments within the standards. Domains included within the CCSSM-C and the grade levels in which they are addressed are:

• Counting and Cardinality (K),
• Operations and Algebraic Thinking (K–5),
• Number and Operations in Base Ten (K–5),
• Measurement and Data (K–5),
• Geometry (K–HS),
• Number and Operations: Fractions (3–5),
• Ratios and Proportional Relationships (6–7),
• The Number System (6–8),
• Expressions and Equations (6–8),
• Statistics and Probability (6–HS),
• Functions (8–HS),
• Number and Quantity (HS),
• Algebra (HS), and
• Modeling (HS).

The CCSSM-P define the process skills that educators need to develop in their students (NGA & CCSSO, 2010a). The following practice standards are for all students in grades kindergarten through college and careers.

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.

To develop innovative and creative mathematicians, Sheffield (2006; Johnsen & Sheffield, 2013) proposed a ninth standard: Solve problems in novel ways and pose new mathematical questions of interest to investigate. In addition to the practice standards, educators should encourage their students to develop a deep understanding of mathematics and aspire to become creative and investigative mathematicians (Johnsen & Sheffield, 2013; Sheffield, 2000, 2003). This outcome can be achieved by (a) having students pose new mathematical questions, add new ideas for solving problems, and create innovative solutions; and (b) having teachers ask questions that encourage mathematical creativity and use assessment criteria that focuses on fluency, flexibility, originality, elaboration or elegance, generalizations, and extensions (Chapin, O’Connor, & Anderson, 2009; Sheffield, 2000).

21st Century Skills

The Mathematical Practice Standards are closely aligned with key 21st century student outcomes (Partnership for 21st Century Skills, n.d.; see Table 1.1). Some of these standards include creativity and innovation, critical thinking and problem solving, and communication and collaboration.

Table 1.1
ComparisonAcrossStandards

Standards for Mathematical Practice	21st Century Skills	NAGC Pre-K-Grade l2 Programming Standards
Students make sense of problems and persevere in solving them. Students solve problems in novel ways and pose new mathematical questions of interest to investigate.	Creativity and innovation.	Educators use creative-thinking strategies (Standard 3.4.2). Educators use inquiry models (Standards 3.4.4).
Students reason abstractly and quantitatively. Students attend to precision. Students look for and make use of structure. Students look for and express regularity.	Critical thinking and problem solving.	Educators use critical-thinking strategies (Standard 3.4.1). Educators use problem-solving model strategies (Standard 3.4.3).
Students construct viable arguments and critique the reasoning of others. Students use appropriate tools strategically.	Communication.	Students develop competence in interpersonal and technical communication skills (Standard 4.5).
Students construct viable arguments and critique the reasoning of others.	Collaboration.	Students possess skills in communicating, teaming, and collaborating with diverse individuals and across diverse groups (Standard 4.4).

Creativity and innovation. Students are encouraged to think creatively using a wide range of ideas, creating new and worth-while ideas, and elaborating and refining their own ideas. They are also expected to work creatively with others by communicating new ideas effectively, being responsive to diverse perspectives, and viewing failure as an opportunity to learn. Moreover, in implementing innovations, students need to learn how to act on creative ideas and make a useful contribution to their field (Partnership for 21st Century Skills, n.d.).

Critical thinking and problem solving. In using these two processes, students need to learn how to reason effectively, use systems thinking to analyze how parts of a whole interact with one another, make judgments and decisions, and solve problems by identifying and asking significant questions that lead to better solutions (Partnership for 21st Century Skills, n.d.).

Communication and collaboration. Students are expected to communicate clearly by articulating their thoughts, listening effectively, using communication for a range of purposes, using and evaluating multiple media and technologies, and communicating effectively in diverse environments. Moreover, they need to learn how to collaborate with others by demonstrating their ability to work effectively and respectfully with diverse teams, exercising flexibility and willingness to be helpful to accomplish a common goal, assuming shared responsibility for collaborative work, and valuing individual contributions (Partnership for 21st Century Skills, n.d.).

Gifted Education Programming Standards

The NAGC Gifted Education Programming Standards (NAGC, 2010) are aligned to the standards of mathematical practice and the 21st century skills (see Table 1.1). For example, gifted and talented students are expected to develop competence in interpersonal and technical communication skills (NAGC, 2010) similar to the 21st century skill of communicating with others and evaluating multiple media. Moreover, teachers of gifted and talented students should use the evidence-based practices of critical thinking, creative thinking, problem solving, and inquiry to encourage students to become independent investigators (NAGC, 2010). Therefore, as educators address one set of national standards, they are addressing others.

Differentiating the Standards

Although the CCSSM standards are strong, they were not developed with the mathematically advanced learner as the focus; therefore, they are not sufficiently advanced to accommodate the needs of most learners who are gifted in mathematics (Johnsen & Sheffield, 2013; VanTassel-Baska, 2013). The CCSSM developers noted that some students may traverse the standards before the end of high school (NGA & CCSSO, 2010b), which will require educators to provide advanced content for them. In addition to subject-based acceleration, educators will also need to enrich the standards by providing more depth, complexity, challenge, and creativity. Educators therefore need a thorough understanding of not only the standards but also the characteristics of gifted and advanced students to differentiate the standards effectively.

Using the CCSSM as a point of departure, we have included differentiation strategies within each of the learning experiences. Educators who teach gifted and advanced students in mathematics might want to use these and other strategies to differentiate the standards (Johnsen & Sheffield, 2013; VanTassel-Baska, 2013).

Acceleration and pacing. Standards and clusters of standards may be identified across grade levels and then compressed within rich problems for more accelerated pacing. Preassessments, curriculum compacting, or above-level testing might be used to determine which students are ready for above-level content (Assouline & Lupkowski-Shoplik, 2011; Colangelo, Assouline, & Gross, 2004). For example, fifth-grade students might be given a problem, such as preparing for the state math assessment, and be required to identify quantitative variables that might improve math performance. Students then might have opportunities to present their data using a variety of graphs; use whole numbers, fractions, or decimals to represent the data; interpret the data using statistical vocabulary; and predict possible trends in student performance. For this book, we decided to select domains that build on one another so that educators can see how to integrate above-level concepts in learning experiences.

Complexity. Complexity can be achieved by solving multi-step, abstract problems at an earlier stage of development or by adding more perspectives or connections (Saul, Assouline, & Sheffield, 2010). For example, in the learning experiences presented in Chapter 4, gifted and advanced third-grade students who are studying the growth of plants might consider multiple variables, such as the amount of pebbles, water, soil, and sunlight instead of measuring only the height of the plant.

Creativity. To enhance creativity and innovation, the teacher might want to develop open-ended problems that offer gifted and advanced learners opportunities to pose their own questions (Sheffield, 2006). These questions may be used to assess not only the number of problems posed but also the complexity of the problems—indices of successful problem solvers (Silver & Cai, 1996). In the learning experiences presented in Chapter 4, problem posing is used at the high school level where students might use census at school data to develop research questions they might want to answer; or, at a kindergarten level, students might sort pictures and/or symbols to compare and contrast their groups using mathematical vocabulary. The point is that educators should be open to removing any grade-level-imposed ceilings on the learner’s opportunities to explore topics.

Depth. In adding depth, the teacher might ask the student for specialized mathematics vocabulary, details, patterns and trends, or rules (Kaplan, 2009). For example, in the learning experiences presented in Chapter 4, gifted and advanced students in mathematics predict their performance on a physical fitness test, collect data, and then check to see if their prediction was confirmed. They might use mathematical vocabulary to describe not only their performance but also the performance of their classmates. Adding this depth accelerates and enriches the learning experience.

Interdisciplinary connections. Because mathematics is a tool subject, it lends itself for interdisciplinary studies (Kaplan, 2009; VanTassel-Baska, 2004). Many examples are available using interdisciplinary research projects in the learning experiences: comparing and contrasting distances and creating a scale map (social studies and math); examining and predicting which variables might influence animal and plant growth (science and math); analyzing data from a physical fitness test (physical education and math); creating a survey to collect information about class-mates’ interests, interpreting the results, and planning meaningful activities (language arts, social studies, and math); and using a mark-recapture experiment (science, math, and engineering).

Themes or concepts. Teachers might have students examine the major concepts or themes within and across domains (VanTassel-Baska, 2004). For example, in the set of experiences presented in Chapter 4, the teacher might ask: What are basic principles in mathematics that are important in studying measurement and data? How might these principles apply to statistics and probability or to another domain, such as number and operati...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Acknowledgments
• Foreword
• Preface
• Chapter 1 Overview
• Chapter 2 Adapting Learning Progressions for Gifted and Advanced Learners
• Chapter 3 Assessment
• Chapter 4 Differentiated Learning Experiences
• Chapter 5 Management Strategies
• Conclusion
• References
• Appendices
• About the Authors