Doing And Talking Mathematics: A Teacher’s Guide To .

8Getting into Position“Doing mathematics” for meaning-making requires that students and teachers reposition themselves: students as questioners and thinkers, andteachers as guides. Both teachers and students may be nervous when these ways of interacting are new. Teachers can help everyone move forwardwhen they steadily support classroom expectations that students should respond to one another’s ideas, that every voice should be heard, that allideas count, that questions are good, and that “wrong” answers help everyone move forward. Over time, students develop experience and increasedconfidence as meaning-makers in mathematics.Helping students become effective and collaborative sense-makers in mathematics means that teachers, like orchestra or choir conductors, set andmaintain a productive tone and rhythm by calling upon different instruments or voices at just the right moment, signaling when each should comein, knowing which should be emphasized, and layering one with another to portray the full complexity of the music (or the math) being conducted.Before a discussion even begins, teachers’ preplanning sets the stage. By selecting high-quality tasks that will give students opportunities to engage indiscourse and use mathematical tools, and planning activities to foster interaction, teachers create a rich context for ect5 Practices for Orchestrating Productive Mathematics Discussions (Smith & Stein, 2011)Through careful preparation, teachers can anticipate the range of student approaches and then monitor them while circulating among studentgroups. They can then select and sequence specific approaches for the whole group to consider. In this way the teacher is able to use evidence ofstudent thinking to build a bridge to the mathematical concepts at the heart of the lesson (Smith & Stein, 2011). It is student reasoning (carefullyprobed and amplified by teachers) that leads to deeper understanding.Asking the Right QuestionsQuestions that provoke productive struggle: big questions. Helping students learn to domathematics for meaning involves helping them develop what Adding It Up (NRC 2001) callsa productive disposition: the tendency to see sense in mathematics, to perceive it as both usefuland worthwhile, to believe that the steady effort in learning mathematics pays off, and to seeThe resources by Leinwand et al and by Smithand Stein, listed under Additional Resources,offer helpful guidance on choosing activities thatpromote deep reasoning.

oneself as an effective learner and doer of mathematics (p. 131). Reaching these big goals requires working with big ideas. If we want to help studentsreason their way through to the mathematical sense at the core of something, it follows that the something must be challenging enough to capture,hold, and be worthy of their attention. Simply stated, the task should connect to a big idea in mathematics, and must present the sort of puzzle forwhich there are multiple entry points and varied solution strategies.Questions that strengthen student reasoning. Understandingthe big idea in mathematics does not happen in one step.Experienced teachers have several sets of purposeful questionsready to help students defend their thinking, look for repeatedreasoning, make generalizations, and prove those generalizations.Facilitating that progression of reasoning takes skill and practice.The Productive Conversation Prompts offer examples of howto press students for meaning and clarity, and how to supporttheir efforts to understand. The Classroom Supports chartoffers a broad array of supports that all students can use as theywork to construct meaning together.Why isthat true?Questions that orchestrate interaction. It will be clear by now that productive discussion isa key component of sense-making in mathematics. Complex knowledge and skills are learnedthrough social interaction as students share ideas, use one another as resources, and togetherconstruct new ways of understanding.Do you think it’salways true?What is yourreasoning?“A classroom is a community of learners.Communities are defined, in part, by how peoplerelate to and interact with each other It mustbe remembered that interacting is not optional:it is essential, because communication isnecessary for building understandings.” Hiebertet al, 1997, p. 9Meaning is not stored language. Meaning is stored experience.Placing students into small groups where they are accountable to one another provides an “Allbrains to work!” opportunity. But students need instruction and support to leverage the powerof shared thinking, since few students come to school knowing how to reason productivelytogether. Some of the Productive Conversation Prompts mentioned above are designed topromote students’ collaborative reasoning. Similarly, students need support in learning howto interact as they build ideas. Issues such as turn-taking, responding in connected, cohesiveways to previous comments, following a chain of reasoning, and challenging someone’s ideawithout challenging the person are all important skills that can be learned and practiced. The9See the bulletin on group work by Lee, Cortada,& Grimm, listed under Additional Resources, forways to support effective group work for EnglishLearners.

10Productive Conversation Moves show seven common interactions as students respondto one another’s ideas, and offer models for these interactions across the range of Englishproficiency. The Conversation Support Card can be used by small groups of students asthey work to build meaning together.Thinking about English Learners: A few modifications to makeChapter 8 in the book by Zwiers, O’Hara, andPritchard, mentioned in the Additional Resourcessection, is an excellent source of advice andpractical strategies for teaching these interactionskills.English Learners have more in common with their English-fluent peers than they havedifferences, but attending to those few differences can help culturally and linguisticallydiverse students become successful meaning-makers with their classmates.Although they sometimes know two or three other languages, English Learners start at different points than their classmates when it comes toEnglish. Those whose proficiency in English is just beginning to develop need ideas presented in multiple ways. Students whose English proficiencyis more developed need fewer and different supports, and are more like their English-fluent peers; with their greater ability to express themselves,they too are learning the vocabulary and language patterns to express new and complex ideas, and to make more precise meanings in mathematics.Their interaction with English-fluent peers as they work out ideas collaboratively provides the impetus to learn new ways of using English to expressideas clearly. The Productive Conversation Moves and Conversation Support Card are all designed to assist English Learners at all levels of Englishproficiency to express their responses to ideas within their groups. The Classroom Supports list many ways teachers can support English Learners asthey do the extra work they must do to create meaning in an English-speaking context, and many more suggestions are readily available. As always,it’s important to think about the principles behind the strategies you use. Here are some that apply to our context:1. Remember three key words: repeated, multiple, and deep. English Learners need repeated exposure to and multiple experiences with ideas inorder to connect the ideas to the English used to express them. They also need information conveyed in multiple ways—voice, print, pictures,activities—so that language alone does not bear the full weight of meaning-making. And English Learners need deep experiences. Rather thanpresent English Learners with new materials to process, it can help to work with the same materials they’ve just encountered, but to do it moredeeply: answer new questions, produce something different, consider the ideas from a different vantage point. As mentioned earlier, meaning isnot stored language; meaning is stored experience, and English Learners will build meaning through multiple related experiences.2. Provide opportunities to discover and discuss the relationship between meaning and language. Questions such as: How do you know that’s whatit’s asking? Where does it say that? What makes you think the writer is uncertain? Where does it show that she is considering a counter-argument? Whatwords tell you that? Which writer is more convincing and why? can help students explore the connection between meaning and language.

3. Discuss the relationship between linguistic choices and the disciplinary purposes it serves. When pressing student to be more precise in theirreasoning (Is it always that way? What if ? How do you know that?), take time to discuss why you are pushing them to be more precise, or moredetailed, or more objective. This provides wonderful opportunities to convey the values of your discipline and to help build the habits of mindstudents will need.Student and teacher positioning. Some English Learners, like some of their classmates, may come from families in which it is not appropriatefor students or children to ask questions or challenge someone’s reasoning. These family views may be supported by cultural patterns that viewknowledge as a commodity in the hands of highly trained and highly educated “others” who are responsible for passing that knowledge on. Whateverits source, this view is worth exploring in the classroom. Explaining to students and families that the teacher’s role is not to give answers, but to teachstudents to think, can shift perceptions. Similarly, if some students have been positioned by their classmates or families as either “poor thinkers” whocan have no worthwhile contributions, or as “brains” who must have all the answers, patience and persistence in helping all students express theirideas and reason collaboratively can help shift those patterns.Meaning-Making in MathematicsMathematics emphasizes using valid, logical reasoning to determine whether a mathematical statement is true or false. Students may begin at thestage of empirical reasoning, giving examples to show that something is likely to be true. At the next stage, informal reasoning, students may be able todescri

mathematics.” (Hiebert et al, 1997, p. 6) More than ever before, mathematics instruction is focused on helping students reason together, guiding students to discover patterns, make sense of mathematical concepts, and make connections between prior knowledge and new learning.