by Laney Sammons

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.

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.

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.

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.

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.

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 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.

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.

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 met them? What do I have to do to meet my goals?

When teachers expect students to share and justify their mathematical reasoning, they provide rich opportunities for self-assessment. By delving more deeply into their thinking to share it with others in mathematical conversations or writing, students often discover gaps in their understanding or lingering questions. Reflecting on their own mathematical understanding related to their learning goals enables students to assume greater responsibility for assessing their own understanding and identifying future learning goals. The extent of their understanding and their need for additional instruction become more visible to them, as well as to their teachers.

The value of this kind of cognitive activity is borne out by research. In his synthesis of over 800 meta-analyses of the influences on student achievement, Hattie (2009) identified the positive impact of visibility in both teaching and learning: "What is most important is that teaching is visible to the student and that the learning is visible to the teacher. The more the student becomes the teacher and the more the teacher becomes the learner, then the more successful are the outcomes" (p. 25). When students are actively engaged in multiple methods of communication, the roles of teacher and learner intermingle. As a result, teaching and learning become more visible to both teachers and learners.

Despite overwhelming agreement as to the value of students mastering both mathematical content and the mathematical processes or practices, classroom instruction too often offers only limited opportunities for students to acquire these skills. While many math resources now include tasks calling for students to engage with both content *and* process, teachers are usually given little guidance on how to teach the mathematical practices themselves. Because communication plays such a large role in these processes, strategies for teaching communication are of particular importance and are the focus of this book.

For many students, learning to communicate effectively about mathematics is comparable to learning a new language. Because they encounter little content-specific mathematical vocabulary in everyday conversations, it is often unfamiliar to them. In fact, Thompson and colleagues (2008) suggest that "we should consider every student a *mathematics language learner* regardless of his or her level of English language proficiency" (p. 11) and provide appropriate language acquisition support. "Just as learning a foreign language is easiest when the learner is thoroughly immersed in the language, the same principle holds true for learning mathematical vocabulary" (Sammons, 2011, p. 60) and for learning how to communicate mathematical thinking effectively.

The acquisition of new languages requires learning how to both comprehend and express ideas. The comprehension of meaning comes from reading and listening. Clearly expressing ideas, on the other hand, requires proficiency in writing and speaking (Chun, 2006). Combined, these skills "are tools for collaboration, discovery, and reflection" (Whitin & Whitin, 2000, p. 2). Effective communicators can both understand the communications of others and express their own ideas so that others understand them. In teaching students to effectively communicate mathematically, teachers need to address reading, listening, writing, and speaking (see Figure 1.1).

Because of the links between language acquisition and the development of mathematical communication skills, math teachers often look to literacy instruction for effective teaching ideas. Literacy strategies are easily adapted for use in teaching math students how to express their ideas more effectively and understand the ideas communicated by others.

Mathematical communication, however, consists of considerably more than just the language components. As it is specific to *mathematical content*, communication is dependent upon a certain degree of mathematical background knowledge. That knowledge is the lens through which a reader or listener makes sense of the words and representations shared by others. Likewise, students need a combination of math content knowledge and the ability to express their ideas precisely when speaking or writing about math so that others can understand them.

It is important for both teachers and students to know what constitutes quality mathematical communication. First, teachers must clarify the criteria for successful performance by students. State standards provide some guidance regarding specific expectations, but to guarantee consistency across grade levels in schools or school districts, it is beneficial for teachers to come together to examine and interpret those standards. By analyzing and assessing samples of student communication relative to the standards, teachers can develop consistent criteria for quality communication to guide both teaching and assessment.

Second, the criteria for success must be shared with students so that they have clear performance targets. Students will only improve their mathematical communication skills if they understand what defines a quality response. The following characteristics of meaningful mathematical communication should be considered when establishing criteria for quality oral and written communication:

- Respectful dialogue with others
- Use of accurate mathematical vocabulary
- Precision in the expression of ideas
- Organized and logical structure
- Use of facts to justify mathematical reasoning
- Participatory listening during conversations
- Careful reading of written mathematical communications
- Comprehension of the oral or written expressions of others
- Requests for clarification, when needed
- Responses that appropriately address the content of communications heard or read
- Disagreement expressed without rancor and supported with evidence
- Maintenance of focus (attentive, staying on topic)

As with any of the mathematical processes or practices, mathematical communication skills are not taught in isolation. Integration of communication into content instruction is crucial to provide authentic contexts. As a part of a content lesson, communication skills may be highlighted in myriad ways:

- Teachers may explicitly teach a communication strategy combined with content in a minilesson or think-aloud.
- Participatory listening skills may be taught and practiced as students listen to their peers describe math concepts or as they brainstorm problem-solving strategies.
- Comprehension strategies may be taught to help students as they decipher word problems.
- Whole-class or small-group lessons may include math conversations focusing on aspects of the content being taught.
- Students may be asked to justify their problem-solving efforts in journals or digital presentations.
- If a class collaboratively collects and displays a set of data graphically, the teacher may lead a shared writing task to communicate the gist of what the data reveals.
- Students might be asked to brainstorm multiple ways to represent a mathematical idea or problem.

It is important to emphasize, however, that there can be no meaningful mathematical communication without mathematical content. Note that these suggested instructional ideas all link to and rely upon the mathematical content being taught.

The chapters that follow present numerous examples of how to encourage and develop students' communication skills in relation to specific math concepts and tasks. Although mathematical tasks in the classroom should always be linked to specific content, there are content-generic prompts, such as those that follow, that can be used to encourage students to communicate their mathematical thinking in a variety of ways and within the context of many kinds of tasks or problems:

- Reflect on your own mathematical thinking about ______. What do you understand? What questions do you have?
- Explain your mathematical thinking to another, either orally or in writing, using representations.
- Justify your own or a peer's problem-solving process.
- Respond to the mathematical ideas of another.
- Explain a mathematical concept or problem so that others will understand it.
- Read and then restate a problem in your own words or represent it in such a way that others will understand it.
- Describe the strategies you used to solve a problem.
- Describe a pattern you notice.
- Make and justify a conjecture based on your observations.
- Share what you wonder about a problem, an observation, or a mathematical idea.
- Explain in detail your mathematical observations.
- Define mathematical terms or concepts in your own words.
- Describe how your math work might be beneficial in other situations.

These are just a few examples of student tasks that provide practice in mathematical communication. Their aim is prompting students to reason, reflect, listen, read, write, speak, represent, justify, observe, and engage more deeply in mathematical thinking as they work to hone their communication skills.

Students of all ages who regularly engage in sharing their mathematical thinking begin to assume a greater responsibility for self-regulating their thinking and their mathematical learning (Chapin, O'Connor, & Anderson, 2003). It encourages them to focus more intently on the mathematical content they are learning, on how well they are learning it, and on what they can do to improve their learning. Establishing an expectation that students express their mathematical thinking orally and in writing is crucial in creating a rich learning environment.

As stated previously, productive student communication "doesn't just occur because the standards demand it or the teacher values it" (Sammons, 2010a, p. 40). An environment in which students productively engage in the sharing of their mathematical thinking is created with intent, offers authentic purposes for communication, and supports learners as they develop a more meaningful understanding of the world of mathematics. There are a number of things that teachers can do to make the climate of their classrooms "communication-friendly" for students:

*Consider how the physical arrangement of the classroom impacts communication.*A classroom in which students are seated in rows of desks tends to inhibit conversation. On the other hand, students who sit so that they face each other can easily engage in sharing their thinking. "Round tables, as opposed to individual desks, offer a gathering place where all come on equal terms. There is no head of the table, no one person elevated above the others" (Nichols, 2006, p. 38).

Additionally, a carpeted gathering area where students can sit together on the floor offers an inviting environment for group conversations. By situating this meeting space adjacent to an interactive whiteboard or chart paper easel, ideas may be recorded to provide a visual representation of the mathematical reasoning being shared or to provide future reference for the class. Making manipulatives accessible during math conversations also allows students to model their mathematical thinking to present it more clearly to others or to explore their own understanding as others share ideas.

To encourage written communication, materials such as journals, graph paper, markers, and common mathematical tools should be available to students. In addition, students can use digital devices to communicate their thinking. Slide shows, blog posts, animations, and recorded oral explanations of thinking are just a few examples of methods now available to students when they communicate mathematically. As with more traditional methods of written communication, making these easily accessible to students encourages their use.*Establish a nonthreatening classroom atmosphere in which errors are viewed as an important part of the learning process.*O'Connell (2007) wisely warns that "there is nothing that will stifle communication more quickly than negative reactions from teachers or other students" (p. 13). Students too often believe that the sole purpose of answering a teacher's question is to give the one and only correct answer. Students are more willing to share their thinking, however, when they understand that their mathematical thinking is highly valued.

Students' comments that reveal misconceptions or errors guide teachers in meeting their instructional needs. If one student has a misconception, it is likely that others share it. By attending carefully to the oral and written communication of their students, teachers can dispel confusion and correct misconceptions in a timely manner. Teachers help students understand this by explicitly explaining why they consider the communication of ideas to be so valuable. When students understand that errors are an integral part of the learning process, they are much more willing to communicate their thinking openly.*Share expectations with students by establishing criteria for quality oral, representational, and written mathematical communication.*Students never arrive at school knowing what constitutes quality mathematical communication; they must be taught. Unless expectations are shared with students, they have little idea of the target they are aiming for. Teachers can offer students guidance regarding their expectations in many ways: modeling and thinking aloud as they communicate, discussing and posting specific criteria for success, providing exemplars of quality communication, and having students examine and analyze examples (videos or written documents) of communication to assess their effectiveness. The chances of students becoming proficient mathematical communicators greatly improve when students are aware of exactly what they are working to achieve.*Create a communication-rich classroom.*Although the focus of this book may be*mathematical*communication, the more students speak, write, and share their thinking in any curricular area, the more adept they become at communicating in general. The benefits of this skill then carry over and enhance mathematical communication.

Cross-curricular communication opportunities abound in self-contained classrooms. In departmentalized math classes, however, more planning may be required. A teacher could post and highlight examples of all kinds of mathematical communication, including newspaper or magazine articles related to math, articles from mathematical blogs, graphic displays of data, mathematical conjectures, or explanations of problem-solving strategies by students. These help students, not only by providing examples they can examine and emulate, but also by making students more aware of the relevance of mathematics in the real world.

When planning lessons, include time daily for discussions of mathematical issues. Support accurate student communication by providing math word walls to reinforce mathematical vocabulary development. Engage students in shared writing activities to create anchor charts and then post them as models of mathematical writing and for future reference by students. Be attentive to examples of mathematical communication that are encountered and search for ways to incorporate them into the classroom.*Provide authentic contexts for mathematical communication by students.*Students are more inclined to think deeply about math and share their thinking when the context is authentic, particularly when their curiosity is aroused about a math-related matter. Literacy author Fox (1993) suggests, "We're currently wasting a lot of time by giving unreal writing tasks in our classrooms … . You and I don't engage in meaningless writing exercises in real life—we're far too busy doing the real thing" (p. 4). In both oral communication tasks and written tasks, students are most engaged when they are "doing the real thing."

Learning is also authentic when students seek to satisfy their own curiosity. Encourage a sense of inquiry in students. Take advantage of teachable moments to spur students' interest in and wonder about the complex discipline of mathematics. Share this wonder with learners. When students are intrigued and see purpose in their communication, their motivation increases, as does the quality of their work.

Designing and implementing an environment that encourages mathematical communication by students is just the first step in teaching students how to share their mathematical ideas effectively. The remaining chapters of this book will address the various forms of mathematical communication, both comprehensive and expressive, and offer specific instructional strategies for helping students become competent communicators. We begin with mathematical conversations—or the skills of speaking and listening—in Chapter 2.

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