Teaching Mathematics in Grades 6 - 12

Developing Research-Based Instructional Practices

Randall E. Groth

• Publisher: SAGE Publications, Inc
• Year: 2012
• ISBN:
  9781483307091
• Edition: 1
• Topic: Pedagogía
• Subtopic: Enseñanza de matemáticas

A journey into the vibrant and intriguing world of mathematics education

Teaching Mathematics in Grades 6 - 12 explores how research in mathematics education can inform teaching practice in grades 6-12. The author shows secondary mathematics teachers the value of being a researcher in the classroom by constantly experimenting with methods for developing students? mathematical thinking and then connecting this research to practices that enhance students? understanding of the material. The chapters in Part I introduce secondary teachers to the field of mathematics education with cross-cutting issues that apply to teaching and learning in all mathematics content areas. The chapters in Part II are devoted to specific mathematics content strands and describe how students think about mathematical concepts. The goal of the text is to have secondary math teachers gain a deeper understanding of the types of mathematical knowledge their students bring to grade 6 – 12 classrooms, and how students? thinking may develop in response to different teaching strategies.

Part I Introduction to Teaching and Learning Mathematics

• Chapter 1 Introduction to Teaching and Learning Mathematics
• Chapter 2 Key Psychological Ideas and Research Findings in Mathematics Education
• Chapter 3 Planning Mathematics Lessons
• Chapter 4 Mathematics Curriculum Models and Techniques
• Chapter 5 Implementing and Assessing Mathematics Lessons and Curricula
• Chapter 6 Becoming a Professional Mathematics Teacher

Chapter 1 Introduction to the Field of Mathematics Education

The past few decades have seen incredible growth in the study of teaching and learning mathematics. K–12 teachers, university professors, and other educators have produced standards documents, research reports, and curriculum frameworks with the potential to help improve students’ learning. All of this activity makes it an exciting time to enter the profession of mathematics teaching. However, it can also be overwhelming to try to digest and reflect on everything the field has to offer. In fact, one is never really done learning about teaching mathematics. The best teachers are always learning ways to improve their practices by talking with colleagues, reading research, reading teachers’ journals, carefully assessing the impact of their instructional practices on their students’ thinking, and adjusting their practices to maximize students’ learning.

The goal of this chapter is to provide a sense of the major issues and trends that have shaped the field of mathematics education in the recent past. By way of introduction, we will examine the standards documents published by the National Council of Teachers of Mathematics (NCTM), an organization with more than 90,000 members dedicated to improving mathematics education. We will then examine trends in mathematics teaching and learning around the world and the central messages of the reform movement in mathematics education. The objective is not to completely “cover” or give a comprehensive treatment of each of these topics—volumes have already been written on each of them—and resources for further study are given at the end of the chapter. Instead, the chapter provides a frame of reference for understanding the rest of the text. Remember, the best teachers are those who are always learning, and reading this chapter represents just the first step in a career-long journey of navigating the field.

A Brief History of Nctm Standards

The 1980s and 1990s marked the beginning of the “standards movement” because of the effort put into developing standards for teaching and learning in various subject areas. NCTM released three standards documents during this period: (1) Curriculum and Evaluation Standards for School Mathematics (1989), (2) Professional Standards for Teaching Mathematics (1991), and (3) Assessment Standards for School Mathematics (1995). The major themes from this first round of standards laid the groundwork for a fourth influential document, Principles and Standards for School Mathematics (NCTM, 2000). To understand the current state of the field of mathematics education, it is important to grasp the central messages conveyed by each document.

Curriculum and Evaluation Standards

NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) described a vision for the teaching and learning of mathematics that differed sharply with much of conventional practice. For example, in regard to algebra, it called for more attention to (1) “developing an understanding of variables, expressions, and equations” (p. 70) and (2) “the use of real-world problems to motivate and apply theory” (p. 126). Less attention was to be given to (1) “manipulating symbols” (p. 70) and (2) “word problems by type, such as coin, digit, and work” (p. 127). The document contained similar direction for other mathematics content areas, including number and operations, geometry, and measurement. The recommendations sought to move school mathematics beyond an exclusive focus on the teaching and learning of procedures. A central emphasis was helping students to understand “the importance of the connections among mathematical topics and those between mathematics and other disciplines” (p. 146).

Curriculum and Evaluation Standards was also revolutionary in its call for more attention to historically neglected areas such as statistics, probability, and discrete mathematics. Although many important applications of these content areas could be found in contemporary society, they were largely absent from the school mathematics curriculum. The recommendation to give more attention to neglected areas was based on the premise that the school curriculum should change as the needs of society change. This premise also dictated that the school curriculum should take advantage of technology to help students understand the conceptual underpinnings of mathematics. In sum, Curriculum and Evaluation Standards recommended reform in what was taught as well as how it was taught.

Professional Standards for Teaching Mathematics

Professional Standards for Teaching Mathematics helped further clarify NCTM's vision for school mathematics reform. It recommended five major shifts in mathematics classroom environments:

• Toward classrooms as mathematical communities—away from classrooms as simply a collection of individuals;
• Toward logic and mathematical evidence as verification—away from the teacher as the sole authority for right answers;
• Toward mathematical reasoning—away from merely memorizing procedures;
• Toward conjecturing, inventing, and problem-solving—away from an emphasis on mechanistic answer-finding;
• Toward connecting mathematics, its ideas, and its applications—away from treating mathematics as a body of isolated concepts and procedures. (NCTM, 1991, p. 3)

NCTM (1991) recognized that these shifts would not occur overnight. Sustained professional development would be necessary to help teachers implement the recommendations.

Stop to Reflect

NCTM's Professional Standards for Teaching Mathematics calls for five major shifts in the environment of mathematics classrooms (see the preceding discussion). What changes, if any, would your past mathematics teachers in Grades K–12 have needed to make to align their instruction with the five recommendations? Provide specific examples.

Assessment Standards for School Mathematics

Assessment Standards for School Mathematics marked the end of the first round of NCTM standards documents. The document defined assessment broadly as “the process of gathering evidence about a student's knowledge of, ability to use, and disposition toward, mathematics and of making inferences from that evidence for a variety of purposes” (NCTM, 1995, p. 3). From this perspective, one of the primary purposes of assessment is to provide teachers information about the nature of student learning. Information about students’ learning can be drawn from a variety of sources. Instead of relying solely on paper-and-pencil tests, teachers can draw information from student interviews, projects, and portfolios. Information gained about students’ learning can in turn help shape future lesson plans.

Principles and Standards for School Mathematics

Principles and Standards for School Mathematics (NCTM, 2000) differed from previous standards documents in that its intent was to write standards that

• build on the foundation of the original Standards documents;
• integrate the classroom-related portions of Curriculum and Evaluation Standards for School Mathematics, Professional Standards for Teaching Mathematics, and Assessment Standards for School Mathematics;
• organize recommendations into four grade bands: prekindergarten through Grade 2, Grades 3–5, Grades 6–8, and Grades 9–12. (p. x)

Principles and Standards for School Mathematics organized its discussion of mathematics content around five content standards: number and operations, algebra, geometry, measurement, and data analysis and probability. The second half of this text uses a similar organizational scheme by devoting chapters to each of the content standards (with the exception that measurement is distributed among the other content strands).

As a consolidation and elaboration of the previous NCTM standards documents, Principles and Standards for School Mathematics represents the closest we have come to a consensus about which mathematical topics should be taught in school and how they should be taught. Teachers, university professors, mathematics supervisors, and other professionals spent three years constructing the document. As it was being written, feedback was elicited from stakeholders in mathematics education around the world. It should be noted, however, that consensus on the vision of NCTM standards has never been, and likely never will be, universal. For example, some disagree with the decreased emphasis on lecture as a teaching method (Wu, 1999b) or the manner in which technology is to be integrated into the curriculum (Askey, 1999). These kinds of conflicts have been characterized as parts of a larger “math war” over the content of the school curriculum (Schoenfeld, 2004). As with most events characterized as “wars,” much of the conflict is based on the opposing sides misunderstanding each other. This book is based on the premise that one must seek to understand the NCTM standards before condemning or accepting them. Toward that end, in the next section, an overview of some of the most important themes in Principles and Standards for School Mathematics is given.

Understanding the Principles and Process Standards

Principles and Standards for School Mathematics goes beyond merely providing an organizational scheme for discussing mathematics content. The document also contains principles and process standards to guide the teaching of mathematics. NCTM principles “describe particular features of high-quality mathematics education” (NCTM, 2000, p. 11), while NCTM process standards describe aspects of mathematical teaching and learning that should occur in all content areas. The six principles are Equity, Curriculum, Teaching, Learning, Assessment, and Technology. The five process standards are Problem Solving, Reasoning and Proof, Communication, Connections, and Representation.

NCTM Principles

The Equity Principle states that “excellence in mathematics education requires equity—high expectations and strong support for all students” (NCTM, 2000, p. 12). This challenges the assumption that only an elite few are meant to understand mathematics. Teachers should expect all students to learn mathematics. The principle calls for equity of access to high-quality mathematics instruction, curriculum materials, and technology. A key to understanding the Equity Principle is the premise that “equity does not mean that every student should receive identical instruction[;] instead, it demands that reasonable and appropriate accommodations be made as needed to promote access and attainment for all students” (NCTM, 2000, p. 12). Therefore, as teachers think about how to achieve equity in their own classrooms, they need to set aside the assumption that “all students should be treated the same way.”

The Curriculum Principle states that “a curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well-articulated across the grades” (NCTM, 2000, p. 14). To understand this principle, it can be helpful to consider a counterexample: some curricula are essentially “laundry lists” of isolated “topics” to be “covered” in a prescribed order. The order in which topics are treated may or may not help students understand the fundamental concepts of the subject. Teachers in such situations may rarely, if ever, plan sequences of instruction with colleagues who teach different grade levels or subjects. The end result is that students perceive mathematics to be a disconnected body of knowledge consisting of isolated rules...