supporting rigorous mathematics teaching and learning

© 2013 UNIVERSITY OF PITTSBURGH © 2013 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Academically Productive Talk in Mathematics: A Means of Making Sense of Mathematical Ideas

Tennessee Department of EducationElementary School MathematicsGrades 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.

© 2013 UNIVERSITY OF PITTSBURGH

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

The Explore Phase/Private Work TimeGenerate Solutions

The Explore Phase/Small Group Problem Solving

1. Generate and Compare Solutions2. Assess and Advance Student Learning

Share, Discuss, and Analyze Phase of the Lesson1. Share and Model

2. Compare Solutions 3. Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write

MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on:• Different solution paths to the same task• Different representations• Errors • Misconceptions

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.

Set Up the TaskSet Up of the Task


Accountable Talk Features 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


Accountable Talk 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 thinkingWhy might this be important?


Accountable Talk Features and IndicatorsAccountability 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 Talk Features and Indicators to Analyze

Classroom Practice


Strings TaskSolve 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 ContextTeacher: Jennifer DiBrienzoGrade: 2School: School #41School 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 IndicatorsAccountability 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

Operations and Algebraic Thinking 2.OARepresent and solve problems involving addition and subtraction.

2.OA.A.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Add and subtract within 20.

2.OA.B.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.


The CCSS for Mathematics: Grade 2

Number and Operations in Base Ten 2.NBTUnderstand place value.

2.NBT.A.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

a. 100 can be thought of as a bundle of ten tens—called a “hundred.”

b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

Common Core State Standards, 2010, p. 19, NGA Center/CCSSO



Number and Operations in Base Ten 2.NBTUnderstand place value.

2.NBT.A.2 Count within 1000; skip-count by 5s, 10s, and 100s.

2.NBT.A.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

2.NBT.A.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.




Number and Operations in Base Ten 2.NBTUse place value understanding and properties of operations to add and subtract.

2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

2.NBT.B.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.



The CCSS for Mathematics: Grade 2Number and Operations in Base Ten 2.NBTUse place value understanding and properties of operations to add and subtract.

2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

2.NBT.B.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

2.NBT.B.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.



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?

1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of

others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO


Essential UnderstandingsReview 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 TaskSolve 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?


Five Representations of Mathematical Ideas

Pictures

Written Symbols

Manipulative

Models

Real-world

Situations

Oral & Written

Language

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 Sou℠ rcebook: 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

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