© 2013 University Of Pittsburgh Supporting Rigorous Mathematics Teaching and Learning Using Assessing and Advancing Questions to Target Essential Understandings Tennessee Department of Education Elementary School Mathematics Grade 4

Rationale There is wide agreement regarding the value of teachers attending to and basing their instructional decisions on the mathematical thinking of their students (Warfield, 2001). By engaging in an analysis of a lesson planning process, teachers will have the opportunity to consider the ways in which the process can be used to help them plan and reflect, both individually and collectively, on instructional activities that are based on student thinking and understanding. 2

© 2013 University Of Pittsburgh Session Goals Participants will: learn to set clear goals for a lesson; learn to write essential understandings and consider the relationship to the CCSS; and learn the importance of essential understandings (EUs) in writing focused advancing questions. 3

© 2013 University Of Pittsburgh Overview of Activities Participants will: engage in a lesson and identify the mathematical goals of the lesson; write essential understandings (EUs) to further articulate a standard; analyze student work to determine where there is evidence of student understanding; and write advancing questions to further student understanding of EUs. 4

TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning The Mathematical Tasks Framework Stein, Smith, Henningsen, & Silver, 2000 Linking to Research/Literature: The QUASAR Project 5

TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning The Mathematical Tasks Framework Stein, Smith, Henningsen, & Silver, 2000 Linking to Research/Literature: The QUASAR Project Setting Goals Selecting Tasks Anticipating Student Responses Orchestrating Productive Discussion Monitoring students as they work Asking assessing and advancing questions Selecting solution paths Sequencing student responses Connecting student responses via Accountable Talk ® discussions Accountable Talk ® is a registered trademark of the University of Pittsburgh 6

© 2013 University Of Pittsburgh Solving and Discussing Solutions to the Thirds and Sixths Task 7

© 2013 University Of Pittsburgh The Structure and Routines of a Lesson The Explore Phase/Private Work Time Generate Solutions The Explore Phase/ Small Group Problem Solving 1.Generate and Compare Solutions 2.Assess and Advance Student Learning 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 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: Engage students in a Quick Write or a discussion of the process. Set Up the Task Set Up of the Task 8

© 2013 University Of Pittsburgh Thirds and Sixths: Task Analysis Solve the task. Write sentences to describe the mathematical relationships that you notice. Anticipate possible student responses to the task. 9

© 2013 University Of Pittsburgh Thirds and Sixths 10

© 2013 University Of Pittsburgh Thirds and Sixths: Task Analysis Study the Grade 4 CCSS for Mathematical Content within the Number and Operations – Fractions domain. Which standards are students expected to demonstrate when solving the fraction task? Identify the CCSS for Mathematical Practice required by the written task. 11

The CCSS for Mathematical Content − Grade 4 Common Core State Standards, 2010, p. 30, NGA Center/CCSSO Number and Operations – Fractions 4.NF Extend understanding of fraction equivalence and ordering. 4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4.NF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 12

The CCSS for Mathematical Content − Grade 4 Common Core State Standards, 2010, p. 30, NGA Center/CCSSO Number and Operations – Fractions 4.NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.B.4a Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). 4.NF.B.4b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) 4.NF.B.4c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? 13

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 14

© 2013 University Of Pittsburgh Common Core Content Standards and Mathematical Practice Standards Essential Understandings The Common Core State Standards 15

Mathematical Essential Understanding (Fractional Equivalence) 4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. ObjectiveEssential Understanding Students will explain with words and diagrams why two fractions are equivalent. When you multiply a fraction by a fraction equivalent to one, the denominator is partitioned into a new designated number of pieces that are smaller in size, but larger in the number of pieces than the original and as a result of the partitioning, all of the pieces referenced by the numerator end up being partitioned, too. 16 Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

Mathematical Essential Understanding (Adding Iterations of a Unit Fraction) 4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. ObjectiveEssential Understanding Students will recognize that adding unit fractions gives you a non-unit fraction. 17 Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

Mathematical Essential Understanding (Multiplying Iterations of a Unit Fraction) 4.NF.B.4a Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). ObjectiveEssential Understanding Students use multiplication of a unit fraction to derive a non-unit fraction. 18 Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

Mathematical Essential Understanding (Equivalent Unit Fraction Expressions) 4.NF.B.4b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) ObjectiveEssential Understanding Students will write equivalent multiplication expressions to represent the same value. 19 Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

Essential Understandings Essential UnderstandingCCSS When creating equivalent fractions, all of the pieces in a whole are subdivided or partitioned, thus the amount of pieces named in the numerator are automatically partitioned in the same way. What is created is an equivalent fraction. 4.NF.A.1 When you multiply a fraction by a fraction equivalent to one, the denominator is partitioned into a new designated number of pieces that are smaller in size, but larger in the number of pieces than the original and as a result of the partitioning of the denominator, all of the pieces referenced by the numerator end up being partitioned, too. 4.NF.A.2 4.NF.B.3 When decomposing a fraction into iterations of the unit fraction, the number of iterations is the same as the value of the numerator. 4.NF.B.4 When “a” identical things are divided into “b” equal parts, each of “a” things contributes 1/b. So, a x 1/b = a/b. (NCTM Essential Understandings, 2011) 4.NF.B.4b 20

© 2013 University Of Pittsburgh Asking Advancing Questions that Target the Essential Understanding 21

© 2013 University Of Pittsburgh Target Mathematical Goal Students’ Mathematical Understandings Assess 22

© 2013 University Of Pittsburgh Target Mathematical Goal A Student’s Current Understanding Advance MathematicalTrajectory 23

© 2013 University Of Pittsburgh Target Target Mathematical Understanding Mathematical Understanding Illuminating Students’ Mathematical Understandings 24

© 2013 University Of Pittsburgh Characteristics of Questions that Support Students’ Exploration Assessing Questions Based closely on the work the student has produced. Clarify what the student has done and what the student understands about what s/he has done. Provide information to the teacher about what the student understands. Advancing Questions Use what students have produced as a basis for making progress toward the target goal. Move students beyond their current thinking by pressing students to extend what they know to a new situation. Press students to think about something they are not currently thinking about. 25

© 2013 University Of Pittsburgh Supporting Students’ Exploration (Analyzing Student Work) Analyze the students’ group work to determine where there is evidence of student understanding. What advancing questions would you ask the students to further their understanding of an EU? 26

Essential Understandings Essential UnderstandingCCSS When creating equivalent fractions, all of the pieces in a whole are subdivided or partitioned, thus the amount of pieces named in the numerator are automatically partitioned in the same way. What is created is an equivalent fraction. 4.NF.A.1 When you multiply a fraction by a fraction equivalent to one, the denominator is partitioned into a new designated number of pieces that are smaller in size, but larger in the number of pieces than the original and as a result of the partitioning of the denominator, all of the pieces referenced by the numerator end up being partitioned, too. 4.NF.A.2 4.NF.B.3 When decomposing a fraction into iterations of the unit fraction, the number of iterations is the same as the value of the numerator. 4.NF.B.4 When “a” identical things are divided into “b” equal parts, each of “a” things contributes 1/b. So, a x 1/b = a/b. (NCTM Essential Understandings, 2011) 4.NF.B.4b 27

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© 2013 University Of Pittsburgh Reflecting on the Use of Essential Understandings How does knowing the essential understandings help you in writing advancing questions? 33