Supporting Rigorous Mathematics Teaching and Learning

Supporting Rigorous Mathematics Teaching and Learning Academically Productive Talk in Mathematics: A Means of Making Sense of Mathematical Ideas Tennessee Department of Education Elementary School Mathematics Grades 1 & 2

Rationale Effective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them (NCTM, 2000). Building a practice of engaging students in academically rigorous tasks supported by Accountable Talk® discourse facilitates effective teaching. Students develop an understanding of mathematical ideas, strategies, and representations; and teachers gain insights into what students know and can do. These insights prepare teachers to consider ways to advance student learning. Today, by analyzing math classroom discussions, teachers will study how Accountable Talk (AT) discussions support student learning and help teachers maintain the cognitive demand of the task. Accountable Talk® is a registered trademark of the University of Pittsburgh

Session Goals Participants will: • learn a set of Accountable Talk features and indicators; and • recognize Accountable Talk stems for each of the features and consider the potential benefit of posting and practicing talk stems with students.

Overview of Activities Participants will: • discuss Accountable Talk features and indicators; • discuss students’ solution paths for a task; • analyze and identify Accountable Talk features and indicators in a lesson; and • plan for an Accountable Talk discussion.

The Structures and Routines of a Lesson Set Up the Task Set Up of the Task MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on: • Different solution paths to the same task • Different representations • Errors • Misconceptions The Explore Phase/Private Work Time Generate Solutions The Explore Phase/Small Group Problem Solving • Generate and Compare Solutions • Assess and Advance Student Learning SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification. REPEAT THE CYCLE FOR EACH SOLUTION PATH COMPARE: Students discuss similarities and difference between solution paths. FOCUS: Discuss the meaning of mathematical ideas in each representation. REFLECT by engaging students in a quick write or a discussion of the process. Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions 3. Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write

Accountable TalkFeatures and Indicators

Accountable Talk Discussion • Study the Accountable Talk features and indicators. • Turn and Talk with your partner about what you would expect teachers and students to be saying during an Accountable Talk discussion for each of the features. • Accountability to the learning community • Accountability to accurate, relevant knowledge • Accountability to discipline-specific standards of rigorous thinking

AccountableTalk Discussion Indicators for all three features must be present in order for the discussion to be an “Accountable Talk Discussion.” • accountability to the learning community • accountability to accurate, relevant knowledge • accountability to discipline-specific standards of rigorous thinking Why might this be important?

Accountable Talk Features and Indicators Accountability to the Learning Community • Active participation in classroom talk. • Listen attentively. • Elaborate and build on each other’s ideas. • Work to clarify or expand a proposition. Accountability to Knowledge • Specific and accurate knowledge. • Appropriate evidence for claims and arguments. • Commitment to getting it right. Accountability to Rigorous Thinking • Synthesize several sources of information. • Construct explanations and test understanding of concepts. • Formulate conjectures and hypotheses. • Employ generally accepted standards of reasoning. • Challenge the quality of evidence and reasoning.

Accountable Talk Starters • Work in triads. • On chart paper, write talk starters for the Accountable Talk indicators. A talk starter is the start of a sentence that you might hear from students if they are holding themselves accountable for using Accountable Talk Moves. e.g., I want to add on to ______. (Community move) The denominator of a fraction tells us _____. (Knowledge move) The two equations are equivalent because ____ (Rigor move). (Work for 5 minutes.)

Accountable Talk Talk Starters What do you notice about the talk starters for the: • accountability to the learning community • accountability to accurate, relevant knowledge • accountability to discipline-specific standards of rigorous thinking What is the distinction between the stems for knowledge and those for rigorous thinking? Why should we pay attention to this?

Using the Accountable TalkFeatures and Indicators to Analyze Classroom Practice

Strings Task Solve the set of addition expressions. Each time you solve a problem, try to use the previous equation to solve the problem. 7 + 3 = ___ 17 + 3 = ___ 27 + 3 = ___ 37 + 3 = ___ 47 + 3 = ___ Solve each problem two different ways. Make a drawing or show your work on a number line. What pattern do you notice? If the pattern continues, what would the next three equations be?

Lesson Context Teacher: Jennifer DiBrienzo Grade: 2 School: School #41 School District: NYC, District 2 Jenniefer DiBrienza is engaging students in solving and discussing the Strings Task. She will engage the class as a whole in discussing the Strings Task and then they will do several problems independently. Jennifer is also showing students how to use a new tool, the open number line.

Reflecting on the Lesson Watch the video. http://tncore.org/sites/www/Uploads/IFL%20Videos/Grade%202%20Videos/2299_DeiBrienza_Strings_new.mp4 What are students learning in the Strings Task? Which Accountable Talk features and indicators were illustrated in the lesson?

Accountable Talk: Features and Indicators Accountability to the Learning Community • Active participation in classroom talk. • Listen attentively. • Elaborate and build on each others’ ideas. • Work to clarify or expand a proposition. Accountability to Knowledge • Specific and accurate knowledge. • Appropriate evidence for claims and arguments. • Commitment to getting it right. Accountability to Rigorous Thinking • Synthesize several sources of information. • Construct explanations and test understanding of concepts. • Formulate conjectures and hypotheses. • Employ generally accepted standards of reasoning. • Challenge the quality of evidence and reasoning.

Thinking Through a Lesson: The Strings Task(Private Think Time and Small Group Time) Work with others at your table. Hold yourselves accountable for engaging in an Accountable Talk discussion when you think through the lesson. • What do students need to understand? • Which solution paths might students use when solving the task? How does one solution path differ from the other? • What questions will you have to ask to address the ideas in the Standards for Mathematical Content?

The CCSS for Mathematical Content: Grade 2 Common Core State Standards, 2010

The CCSS for Mathematics: Grade 2 Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

Common Core State Standards for Mathematical Practice What would have to happen in order for students to have opportunities to make use of the CCSS for Mathematical Practice? • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning. Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO

Essential Understandings Review the essential understandings for the lesson. Which essential understandings will students be left with after the lesson? • Counting strategies are based on order and hierarchical inclusion of numbers. (NCTM) • Counting includes one-to-one correspondence, regardless of the kind of objects in the set and the order in which they are counted. (NCTM) • Counting tells how many items there are altogether. When counting, the last number tells the total number of items. (NCSM Journal/Van de Walle spring-summer 2012) • Sets of ten (and tens of tens) can be perceived as single entities. These sets can then be counted and used as a means of describing quantities. • The positions of digits in numbers determine what they represent–which size group they count. This is the major principle of place value numeration. • There are patterns to the way that numbers are formed. For example, each decade has a symbolic pattern reflective of the 1-9 sequence. • The groupings of ones, tens, and hundreds can be taken apart in different ways. For example, 256 can be 1 hundred, 14 tens, and 16 ones but also 250 and 6. Taking numbers apart and recombining them in flexible ways is a significant skill for computation.

Strings Task Solve the set of addition expressions. Each time you solve a problem, try to use the previous equation to solve the problem. 7 + 3 = ___ 17 + 3 = ___ 27 + 3 = ___ 37 + 3 = ___ 47 + 3 = ___ Solve each problem two different ways. Make a drawing or show your work on a number line. What pattern do you notice? If the pattern continues, what would the next three equations be?

Reflection on the LessonCommon Core State Standards (CCSS) Examine the second grade CCSS for Mathematics. • Which CCSS for Mathematical Content did we discuss? • Which CCSS for Mathematical Practice did we use when solving and discussing the task?

Pictures Manipulative Models Written Symbols Real-world Situations Oral & Written Language Five Representations of Mathematical Ideas Modified from Van De Walle, 2004, p. 30

Accountable Talk Discussion Successful teachers are skillful in building shared contexts of the mind (not merely assuming them) and assuring that there is equity and access to these experiences. Talk about these experiences for all members of the classroom are a necessary part of the experience. Over time, these contexts of the mind and collective experiences with talk lead to the development of a "discourse community"—with shared understandings, ways of speaking, and new discursive tools with which to explore and generate knowledge. In this way, an intellectual "commonwealth" can be built on a base of tremendous sociocultural diversity. Accountable Talk℠ Sourcebook: For Classroom Conversation that Works (IFL, 2010)

Reflection • What will you keep in mind when attempting to engage students in Accountable Talk discussions? • What does it take to maintain the demands of a cognitively demanding task during the lesson so that you have a rigorous mathematics lesson? • What role does talk play?

Reflecting on the Accountable Talk Discussion Step back from the discussion. What are some patterns that you notice? What mathematical ideas does the teacher want students to discover and discuss?

Bridge to Practice • Plan a lesson with colleagues. Select a high-level task. • Anticipate student responses. Discuss ways in which you will engage students in talk that is accountable to community, to knowledge, and to standards of rigorous thinking. Specifically, list the moves and the questions that you will ask during the lesson. • Engage students in an Accountable Talk discussion. Ask a colleague to scribe a segment of your lesson, or audio or video tape your own lesson and transcribe it later. • Analyze the Accountable Talk discussion in the transcribed segment of the talk. Identify talk moves and the purpose that the moves served in the lesson.

Supporting Rigorous Mathematics Teaching and Learning