Teaching Students to Communicate Mathematically

Laney Sammons

• Publisher: ASCD
• Year: 2018
• ISBN:
  9781416625605
• Topic: Bildung
• Subtopic: Bildung Allgemein

Students learning math are expected to do more than just solve problems; they must also be able to demonstrate their thinking and share their ideas, both orally and in writing. As many classroom teachers have discovered, these can be challenging tasks for students. The good news is, mathematical communication can be taught and mastered.

In Teaching Students to Communicate Mathematically, Laney Sammons provides practical assistance for K–8 classroom teachers. Drawing on her vast knowledge and experience as a classroom teacher, she covers the basics of effective mathematical communication and offers specific strategies for teaching students how to speak and write about math. Sammons also presents useful suggestions for helping students incorporate correct vocabulary and appropriate representations when presenting their mathematical ideas.

This must-have resource will help you help your students improve their understanding of and their skill and confidence in mathematical communication.

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Chapter 1

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The Essentials of Mathematical Communication

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Communication is an essential part of mathematics and mathematics education. It is a way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment. The communication process also helps build meaning and permanence for ideas and makes them public (National Council of Teachers of Mathematics [NCTM], 2000, p. 60).

In mathematics, as in all subject areas, the true essence of teaching is guiding others to greater understanding. Exemplary math teachers nurture their students' appreciation of the discipline and lead them to an understanding of math that can be applied in diverse situations. This kind of teaching does more than simply impart facts and procedures that are devoid of context or meaning; it taps into the curiosity of learners and offers them opportunities for mathematical exploration, with teachers and learners working collaboratively to construct knowledge (Mercer, 1995). Essential to this learning process is effective communication.

What Is Mathematical Communication—and Why Write a Book About It?

Merriam-Webster (2017) defines communication as "a process by which information is exchanged between individuals through a common system of symbols, signs, or behavior." As such, mathematical communication entails a wide range of cognitive skills. Because it is an exchange of ideas, it encompasses both listening and reading (comprehension) and both speaking and writing (expression). Somewhat unique to math, expression may also include the representation of mathematical ideas in nonlinguistic ways.

Whereas math teachers have traditionally focused on teaching content, more challenging standards are encouraging educators to expand their instruction to promote students' mathematical practice skills, most of which depend heavily on learning to communicate effectively about math. As a later summary of current math practice standards will show, most educators agree that effective communication is critical for more rigorous instruction and deeper mathematics learning, and many teaching materials and resources now include tasks that require it. Too often, however, students receive little or no instruction on how to communicate about math effectively before they are asked to do so. Naturally, this sets them up for failure. Just requiring students to justify their reasoning does not work; they must know what makes math talk and math writing effective. Students should be explicitly taught these essential skills and given ample opportunities to practice independently, beginning in kindergarten.

This book is designed to provide educators with strategies for teaching students to express their mathematical thinking effectively—orally, with the use of representations, and in writing. Successful literacy strategies, including word walls, modeling, shared writing and revision, and exemplars—strategies that show students how to talk about and write about math, rather than only assigning tasks that require it without having first taught students how to do it—will be closely examined.

After establishing what mathematical communication is and why it is essential, Chapters 2–7 examine the individual components of this type of communication. In addition to the components that are most commonly the focus of teaching mathematical communication—math vocabulary, discourse, and writing—representation as a method of communicating mathematical thinking is highlighted. Sample lessons and classroom scenarios for grade level bands K–2 and 3–5 and upper grades are included throughout, with specific instructional ideas you can use with your students.

As will become clear, the benefits of engaging students in mathematical communication go far beyond helping students meet required standards or achieve higher grades. Simply by going through the process of reflecting, organizing their thoughts, and deciding how to express those thoughts in words, students learn to think more deeply, assess their own understanding, make connections, determine importance, and compare ideas. The ongoing interaction with mathematical vocabulary helps reinforce students' understanding, not only of the words themselves, but also of the mathematical ideas the words express. The teaching ideas and examples in this book are offered as a path to more rigorous instruction in which students are immersed, with the help of effective communication, in the fascinating and challenging discipline of mathematics.

Mathematical Communication and the Standards

Most mathematics standards now address content and process. While the mathematical content for grade levels varies and builds from year to year, the processes or practices remain more consistent. They specify the ways students should learn to interact with math—how students learn to act as true mathematicians. In examining the processes and practices prescribed in these standards, the importance of teaching students how to communicate mathematically is clear. It's worth taking a brief look at how communication is treated in the documents that currently guide instructional goals and curriculum development for K–12 mathematical instruction: the National Council of Teachers of Mathematics (NCTM) Process Standards (2000), the Common Core State Standards for Mathematical Practice (2010), and selected state standards that are largely based on the first two.

National Council of Teachers of Mathematics Process Standards

In 2000, the NCTM introduced a set of principles and standards for mathematics instruction that include both content and process standards. While proficiency in mathematical communication is implicit in many of the content standards, it is inherent in the NCTM process standards described here.

Problem solving. Students should be engaged in solving problems posed in math class, as well as those that occur in real-life situations. They should be encouraged to construct new mathematical meaning from their problem-solving efforts. Being able to communicate mathematically is essential for these tasks. First, students must make sense of problems, make connections to the math they know, and then translate the problems into mathematical terms. According to the NCTM (2000), good problem solvers "monitor and reflect on the process of mathematical problem solving" (p. 52) and adjust their use of strategies as needed. "Such reflective skills are much more likely to develop in a classroom environment that supports them" (p. 54). This standard requires that teachers establish a learning environment in which students develop the habit of reflection through conversation, beginning in the early grades.

Reasoning and proof. Students should understand that reasoning and proof are fundamental to the discipline of mathematics. As learners "make and investigate conjectures" or "develop and evaluate mathematical arguments and proofs" (p. 56), strong communication skills are essential.

Communication. This standard is explicit in emphasizing the importance of students being able to "organize and consolidate their thinking through communication," as well as being able to "communicate their mathematical thinking coherently and clearly to their peers, teachers, and others" (p. 60). They must also "analyze and evaluate the mathematical thinking and strategies of others" and "use the language of mathematics to express mathematical ideas precisely" (p. 60).

Connections. Learners should "recognize and use connections among mathematical ideas" and "understand how mathematical ideas interconnect and build upon one another to produce a coherent whole" (p. 64). They should also be able to "recognize and apply mathematics in context outside of mathematics" (p. 64). Communicating their ideas is valuable in leading students to clarify and organize their thinking more effectively and to help them recognize important mathematical connections.

Representation. This standard specifically states that students create and use representations to "organize, record, and communicate mathematical ideas" (p. 67), underscoring the crucial role of communication in mathematical proficiency.

Common Core State Math Standards of Mathematical Practice

The Common Core State Standards include eight Standards for Mathematical Practice that apply to students from kindergarten to 12th grade. Briefly, students should be able to perform the following important tasks:

• Make sense of problems,
• Reason abstractly,
• Construct arguments and critique the reasoning of others,
• Construct mathematical models,
• Use appropriate tools,
• Attend to precision,
• Make use of structure, and
• Look for and express regularity in repeated reasoning.

In examining these tasks, it is obvious that communication is key to many of them. To construct mathematical models, students must construct representations of mathematical thinking—a crucial element of communication. As well, to construct arguments, critique the reasoning of others, attend to precision, or express regularity in repeated reasoning, students must be able to clearly communicate their mathematical thinking.

Standards in Various States

Various states have adopted their own math standards that address specific processes or practices. For example, the Texas Essential Knowledge and Skills for Mathematics (2012) highlight the process standards and their important relationship to mathematical content instruction with this explanation:

The process standards describe ways in which students are expected to engage in the content. The placement of the process standards at the beginning of the knowledge and skills listed for each grade and course is intentional. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. The process standards are integrated at every grade level and course.

In some states, while there may be no specific process or practice standards, they are still considered essential for mathematical proficiency. The Mathematics Standards of Learning for Virginia Public Schools, for example, establish five goals for mathematics instruction:

• Becoming mathematical problem solvers,
• Communicating mathematically,
• Reasoning mathematically,
• Making mathematical connections, and
• Using mathematical representations to model and interpret practical situations (Board of Education of the Commonwealth of Virginia, 2016, p. v).

The Nebraska Department of Education (2015) also describes four mathematical processes—problem solving, modeling and representation, communication, and making connections—stating that they "reflect the interaction of skills necessary for success in math coursework, as well as the ability to apply math knowledge and processes within real-world contexts" (p. 2).

The Importance of Teaching Mathematical Communication

The individuals who develop math standards recognize the importance of teaching students of all grade levels how to communicate mathematically. Incorporating ongoing opportunities for mathematical communication (oral, representational, or written) as an integral part of instruction not only enhances student learning, but also provides students with much-needed life skills. To avoid what Wagner (2008) calls a "global achievement gap," schools are working to extend instruction beyond the simple acquisition of knowledge to reflect the demands of life in the 21st century. Educators are helping students learn how to think critically, work collaboratively with their peers, access and accurately analyze relevant information, and solve problems effectively. Because the ability to communicate is crucial for these important life skills, creating classroom environments in which students regularly practice multiple forms of communication is imperative.

Building Mathematical Comprehension

As noted, the standards incorporate mathematical communication as an essential competency in and of itself—something required of mathematicians and necessary for meeting the everyday demands of life in our society. However, participating in oral and written communication also enhances students' conceptual understanding of mathematics; by melding their own ideas with those of others, their mathematical understanding is refined and expanded.

Furthermore, thoughtful communication requires careful reasoning. As Chapin and colleagues (2003) pointed out, "We reason when we examine patterns and detect regularities, generalize relationships, make conjectures, and evaluate or construct an argument" (p. 79). "Being asked, ‘Why do you think that?’ has profound effects on students' mathematical comprehension and on their ‘habits of mind’ in general" (p. 19).

Students preparing to share their thinking must incorporate the following activities:

• Review what they know about math related to the topic,
• Make mathematical connections,
• Organize their ideas,
• Determine the relative importance of their ideas to the math topic,
• Decide which ideas to share with others,
• Identify the appropriate mathematical vocabulary terms to use when communicating their ideas,
• Compose a statement that clearly explains their ideas, and then
• Express their thinking orally, with representations, and/or in writing.

Additionally, students must use the following skills in conversations or when reading other students' written communications:

• Listen to or read others' mathematical ideas,
• Compare those ideas to what they already know and think,
• Construct new knowledge or meaning by melding the new ideas with their own thinking,
• Decide what thoughts to include in a response,
• Compose a response, and
• Deliver the response.

The entire process begins anew when students listen to or read responses from other students. When that happens, students must attend to any feedback offered by peers or teachers and then cycle back through the steps of this thinking process.

With each step, students revisit and reconsider relevant mathematical ideas, often in a new light. By touching on these mathematical ideas repeatedly while communicating, students' mathematical understanding broadens and deepens. As a result, fledgling mathematicians can apply what they are learning and are more likely to retain their new knowledge and skills.

With many opportunities to communicate their thinking, the mathematical understanding of students grows deeper and becomes more complex. Students learn "the power that comes from wrestling with an inkling of an idea, shaping and articulating it the best they can, then working with others to enable their idea to gain strength and grow" (Nichols, 2006, p. 33). This is mathematical learning at its best, going well beyond the simple acquisition of facts, procedural knowledge, and computational fluency.

Making Mathematical Thinking Visible

The critical role of formative assessment in both teaching and learning has been well documented (Black & Wiliam, 2010; Fisher & Frey, 2007; Stiggins, 1997, 2002, 2005). Listening to and reading about students' explanations of their mathematical thinking offers teachers an accurate assessment of students' knowledge and skills and helps them target specific learning needs. It is this "kind of thoughtful practice that drives effective teaching" (Rowan & Bourne, 2001, p. 37). In a sense, students begin to think as teachers: What should I know and be able to do? What are my learning goals? Have I m...

Table of contents

• Cover
• Title Page
• Table of Contents
• Chapter 1. The Essentials of Mathematical Communication
• Chapter 2. Effective Mathematical Conversations
• Chapter 3. Teaching Students to Engage in Mathematical Conversations
• Chapter 4. Writing About Math
• Chapter 5. Teaching Students to Write About Mathematics
• Chapter 6. Developing Mathematical Vocabulary Knowledge
• Chapter 7. Mathematical Representations
• References
• Study Guide
• Related ASCD Resources
• About the Author
• Copyright