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Supporting Rigorous Mathematics Teaching and Learning

Engaging In and Analyzing Teaching and Learning

Tennessee Department of EducationElementary School MathematicsGrade 2

RationaleAsking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true - Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

By engaging in a task, teachers will have the opportunity to consider the potential of the task and engagement in the task for helping learners develop the facility for expressing a relationship between quantities in different representational forms, and for making connections between those forms.

Common Core State Standards for Mathematics, 2010

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Session Goals

Participants will:

• develop a shared understanding of teaching and learning; and

• deepen content and pedagogical knowledge of mathematics as it relates to the Common Core State Standards (CCSS) for Mathematics.

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Overview of Activities

Participants will:

• engage in a lesson; and• reflect on learning in relationship to the CCSS.

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Looking Over the Standards

• Read the task. Before you solve the task, look over the second grade standards for Operations and Algebraic Thinking and Number Operations in Base Ten.

• We will return to the standards at the end of the lesson and consider what it means to say:

In what ways did we have opportunities to learn about the concepts underlying the standards?

What gets “counted” as learning?

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Eduardo’s and Katrina’s Strategies

Eduardo solves the story problem below by using subtraction. Show Eduardo’s equation.

You have 100 stickers. You put 48 of the stickers into your sticker album. How many stickers do you still need to put in an album?

When Eduardo compares his work to Katrina’s, he sees that she used addition to solve the problem.

Explain to Eduardo why Katrina can use addition to solve this problem.

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The CCSS for Mathematics: Grade 2Operations 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.

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

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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 8

The CCSS for Mathematics: Grade 2Number 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.

Common Core State Standards, 2010, p. 19, NGA Center/CCSSO 9

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.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.

Common Core State Standards, 2010, p. 19, NGA Center/CCSSO 10

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, 2010, p. 19, NGA Center/CCSSO 11

Table 1: Common Addition and Subtraction Situations

Common Core State Standards, 2010 12

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Engage In and Reflect on a Lesson

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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

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Solve the Task(Private Think Time)

• Work privately on the Eduardo’s and Katrina’s Strategies Task.

• Work with others at your table. Compare your solution paths.

• Make observations about relationships that you notice.

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Eduardo’s and Katrina’s Strategies

Eduardo solves the story problem below by using subtraction. Show Eduardo’s equation.

You have 100 stickers. You put 48 of the stickers into your sticker album. How many stickers do you still need to put in an album?

When Eduardo compares his work to Katrina’s, he sees that she used addition to solve the problem.

Explain to Eduardo why Katrina can use addition to solve this problem.

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Expectations for Group Discussion

• Solution paths will be shared.

• Listen with the goals of:– putting the ideas into your own words;– adding on to the ideas of others;– making connections between solution paths;

and– asking questions about the ideas shared.

• The goal is to understand the mathematical relationships and to make connections among the various strategies used when solving the problems in the task.

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Reflecting on Our Learning

• What supported your learning?

• Which of the supports listed will EL students benefit from during instruction?

• Which CCSS for Mathematical Content did we discuss?

• Which CCSS for Mathematical Practice did you use when solving the task?

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Linking to Research/LiteratureConnections Between Representations

Adapted from Lesh, Post, & Behr, 1987

Pictures

WrittenSymbols

ManipulativeModels

Real-worldSituations

Oral Language

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Reflecting on Our Learning

• What supported your learning?

• Which of the supports listed will EL students benefit from during instruction?

• Which CCSS for Mathematical Content did we discuss?

• Which CCSS for Mathematical Practice did you use when solving the task?

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© 2013 UNIVERSITY OF PITTSBURGH

Reflecting on Our Learning

• What supported your learning?

• Which of the supports listed will EL students benefit from during instruction?

• Which CCSS for Mathematical Content did we discuss?

• Which CCSS for Mathematical Practice did you use when solving the task?

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The CCSS for Mathematical Practice1. 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

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Research Connection: Findings from Tharp and Gallimore

• For teaching to have occurred - Teachers must “be aware of the students’ ever-changing relationships to the subject matter.”

• They [teachers] can assist because, while the learning process is alive and unfolding, they see and feel the students’ progression through the zone, as well as the stumbles and errors that call for support.

• For the development of thinking skills—the [students’] ability to form, express, and exchange ideas in speech and writing—the critical form of assisting learners is dialogue—the questioning and sharing of ideas and knowledge that happen in conversation.

Tharp & Gallimore, 1991

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