Principles to Actions Effective Mathematics Teaching Practices The

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Principles to Actions Effective Mathematics Teaching Practices The Case of Katherine Casey and the

Principles to Actions Effective Mathematics Teaching Practices The Case of Katherine Casey and the Multiplication Strings Task Grade 4 This module was developed by De. Ann Huinker, University of Wisconsin-Milwaukee; Victoria Bill, University of Pittsburgh Institute for Learning; and Amy Hillen, Kennesaw State University. Video courtesy of New York City Public Schools and the University of Pittsburgh Institute for Learning. These materials are part of the Principles to Actions Professional Learning Toolkit: Teaching and Learning created by the NCTM project team that includes: Margaret Smith (chair), Victoria Bill (co-chair), Melissa Boston, Fredrick Dillon, Amy Hillen, De. Ann Huinker, Stephen Miller, Lynn Raith, and Michael Steele.

Overview of the Session • Watch a video clip of a fourth grade class

Overview of the Session • Watch a video clip of a fourth grade class engaged in a whole class discussion of a series of related multiplication equations. • Discuss what the teacher does to support the students’ learning of mathematics. • Relate teacher actions in the video to the Effective Mathematics Teaching Practices.

Teaching and Learning Principle An excellent mathematics program requires effective teaching that engages students

Teaching and Learning Principle An excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their ability to make sense of mathematical ideas and reason mathematically. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 7)

Ms. Casey’s Fourth Grade Classroom “Multiplication Strings Task”

Ms. Casey’s Fourth Grade Classroom “Multiplication Strings Task”

Multiplication Strings Task (Teacher Version) 1. Solve this set of multiplication equations. Each time

Multiplication Strings Task (Teacher Version) 1. Solve this set of multiplication equations. Each time you solve a new equation, try to use the previous equation to help you solve it. 8 x 4 = ____ 8 x 8 = ____ 8 x 16 = ____ 8 x 32 = ____ 8 x 64 = ____ 2. Reflect on the string of multiplication equations. What patterns do you notice? Why do these patterns occur? 3. To solve 8 x 8, Nicholas, a third grade student, explained that he knew that 8 times 4 would be 32 so he just added 32 and 32 to get the answer. Show you, as the teacher, might represent his reasoning with an area model. How does his strategy utilize properties of the operations?

Ms. Casey’s Mathematics Learning Goals Students will understand that: • Multiplication can be represented

Ms. Casey’s Mathematics Learning Goals Students will understand that: • Multiplication can be represented by the area of a rectangle because tiling the figure would show rows with an equal number or columns with equal number of square units. • Decomposing and recomposing groups, based on properties of the operations, makes it possible to flexibly and fluently make use of known multiplication combinations to solve unknown multiplication equations. • Systematically analyzing what happens to the product when one factor is changed reveals patterns and regularity in repeated reasoning that can lead to flexible and fluent multiplication strategies.

Connections to CCSSM Grade 3 Standards for Mathematical Content Understand properties of multiplication and

Connections to CCSSM Grade 3 Standards for Mathematical Content Understand properties of multiplication and the relationship between multiplication and division. 3. OA. B. 5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication. ) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication. ) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) =(8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property. ) Solve problems involving the four operations; identify and explain patterns in arithmetic. 3. OA. D. 9 Identify arithmetic patterns (including patterns in the addition table or multiplication table). And explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. Relate area to the operations of multiplication and addition. 3. MD. C. 7 c Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a x b and a x c. Use area models to represent the distributive property in mathematical reasoning. 3. MD. C. 7 d Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). (2014). Common core state standards for mathematics. Retrieved from http: //www. corestandards. org/Math/

Connections to CCSSM Grade 4 Standards for Mathematical Content Use place value understanding and

Connections to CCSSM Grade 4 Standards for Mathematical Content Use place value understanding and properties of operations to perform multi-digit arithmetic. 4. NBT. B. 5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). (2014). Common core state standards for mathematics. Retrieved from http: //www. corestandards. org/Math/

Connections to the CCSSM Standards for Mathematical Practice 1. Make sense of problems and

Connections to the CCSSM Standards 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. National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). (2014). Mathematics. Common core state standards for mathematics. Retrieved from http: //www. corestandards. org/Math/Practice

Standards for Mathematical Practice (SMP) SMP 7. Look for and make use of structure.

Standards for Mathematical Practice (SMP) SMP 7. Look for and make use of structure. Mathematically proficient students at the elementary grades use structures such as place value, the properties of operations, other generalizations about the behavior of the operations, and attributes of shapes to solve problems. In many cases, they have identified and described these structures through repeated reasoning (SMP 8). When older students calculate 16 x 9, they might apply the structure of place value and the distributive property to find the product: 16 x 9 = (10 + 6) x 9 = (10 x 9) + (6 x 9). To determine the volume of a 3 x 4 x 5 rectangular prism, students might see the structure of the prism as five layers of 3 x 4 arrays of cubes. Illustrative Mathematics. (2014, February 12). Standards for Mathematical Practice: Commentary and Elaborations for K– 5. Tucson, AZ. . Retrieved from http: //commoncoretools. me/wp-content/uploads/2014/02/Elaborations. pdf (pp. 18 -19)

Standards for Mathematical Practice (SMP) SMP 8. Look for and express regularity in repeated

Standards for Mathematical Practice (SMP) SMP 8. Look for and express regularity in repeated reasoning. Mathematically proficient students at the elementary grades look for regularities as they solve multiple related problems, then identify and describe these regularities. For example, students might notice a pattern in the change to the product when a factor is increased by 1: 5 x 7 = 35 and 5 x 8 = 40, the product changes by 5; 9 x 4 = 36 and 10 x 4 = 40, the product changes by 4. Students might then express this regularity by saying something like, “When you change one factor by 1, the product increases by the other factor. ” Mathematically proficient students formulate conjectures about what they notice. . As students practice articulating their observations, they learn to communicate with greater precision (SMP 6). As they explain why these generalizations must be true, they construct, critique, and compare arguments (SMP 3).

Classroom Context for the Video Segment Teacher: Katherine Casey Grade: 4 School: PS 116

Classroom Context for the Video Segment Teacher: Katherine Casey Grade: 4 School: PS 116 District: New York Community School District 2 Month: November Number of students in the classroom: 27 The class has been creating all of the possible rectangular arrays for a given number of square units. In this video, the teacher poses a sequence of equations for the students to solve. Students are challenged to use one equation to solve another equation. The students share their strategies and discuss “what they notice” about the multiplication equations and the products. The teacher uses an area model to represent the students’ reasoning.

Rectangular Arrays rot a te 8 x 4 8 rows with 4 in each

Rectangular Arrays rot a te 8 x 4 8 rows with 4 in each row 4 columns with 8 in each column 4 x 8 4 rows with 8 in each row 8 columns with 4 in each column Area Models 4 cm rot a te 8 cm 4 cm

 Multiplication Strings: Board Work 4, 8, 12, 16, 20, 24, 28, 32 8

Multiplication Strings: Board Work 4, 8, 12, 16, 20, 24, 28, 32 8 + 8 = 16 8 x 4 = 32 x 2 16 + 16 = 32 8 x 8 = 64 x 2 8 x 2 4 4 8 32 32 = 64 16 8 x 16 = 128 x 2 8 x 32 = 256 x 2 8 8 8 64 64 = 128

Lens for Watching the Video As you watch the video • Identify the mathematical

Lens for Watching the Video As you watch the video • Identify the mathematical insights that surfaced for students, and • Make note of what the teacher does to support student learning of mathematics.

Watch the Video “Multiplication Strings”

Watch the Video “Multiplication Strings”

Discussing Student Learning of Mathematics Turn and Talk: 1. What mathematical insights surfaced for

Discussing Student Learning of Mathematics Turn and Talk: 1. What mathematical insights surfaced for students? 2. In what ways did the teacher advance student understanding toward the learning goals for the lesson?

A Look at the Effective Mathematics Teaching Practices

A Look at the Effective Mathematics Teaching Practices

Effective Mathematics Teaching Practices 1. Establish mathematics goals to focus learning. 2. Implement tasks

Effective Mathematics Teaching Practices 1. Establish mathematics goals to focus learning. 2. Implement tasks that promote reasoning and problem solving. 3. Use and connect mathematical representations. 4. Facilitate meaningful mathematical discourse. 5. Pose purposeful questions. 6. Build procedural fluency from conceptual understanding. 7. Support productive struggle in learning mathematics. 8. Elicit and use evidence of student thinking. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

Teacher Actions As you reflect on the teacher actions that supported student learning in

Teacher Actions As you reflect on the teacher actions that supported student learning in this lesson, which Effective Mathematics Teaching Practices did you notice the teacher using?

Effective Mathematics Teaching Practice Build Procedural Fluency from Conceptual Understanding

Effective Mathematics Teaching Practice Build Procedural Fluency from Conceptual Understanding

Build Procedural Fluency from Conceptual Understanding Effective teaching of mathematics • builds on a

Build Procedural Fluency from Conceptual Understanding Effective teaching of mathematics • builds on a foundation of conceptual understanding; • results in generalized methods for solving problems; and • enables students to flexibly choose among methods to solve contextual and mathematical problems. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 42)

Teaching Practice Focus: Build Procedural Fluency from Conceptual Understanding Jot down your responses. •

Teaching Practice Focus: Build Procedural Fluency from Conceptual Understanding Jot down your responses. • What does it mean to be fluent with computational procedures? • Why is it important to build procedures from conceptual understanding?

Procedural Fluency Being fluent means that students are able to choose flexibly among methods

Procedural Fluency Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems, they understand are able to explain their approaches, and they are able to produce accurate answers efficiently. Fluency builds from initial exploration and discussion of number concepts to using informal reasoning strategies based on meanings and properties of the operations to the eventual use of general methods as tools in solving problems. This sequence is beneficial whether students are building toward fluency with single- and multi-digit computation with whole numbers or fluency with, for example, fraction operations, proportional relationships, measurement formulas, or algebraic procedures. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 42)

Conceptual Understanding & Procedural Fluency When procedures are connected with the underlying concepts, students

Conceptual Understanding & Procedural Fluency When procedures are connected with the underlying concepts, students have better retention of the procedures and are more able to apply them in new situations (Fuson, Kalchman, and Bransford 2005). Martin (2009, p. 165) describes some of the reasons that fluency depends on and extends from conceptual understanding: To use mathematics effectively, students must be able to do much more than carry out mathematical procedures. They must know which procedure is appropriate and most productive in a given situation, what a procedure accomplishes, and what kind of results to expect. Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 42)

Teacher and Student Actions: Build Procedural Fluency from Conceptual Understanding What are teachers doing?

Teacher and Student Actions: Build Procedural Fluency from Conceptual Understanding What are teachers doing? What are students doing? • Providing students with opportunities to use their own reasoning strategies and methods for solving problems. • Asking students to discuss and explain why the procedures that they are using work to solve particular problems. • Connecting student-generated strategies and methods to more efficient procedures as appropriate. • Using visual models to support students’ understanding of general methods. • Providing students with opportunities for distributed practice of procedures. • Making sure that they understand can explain the mathematical basis for the procedures that they are using. • Demonstrating flexible use of strategies and methods while reflecting on which procedures seem to work best for specific types of problems. • Determining whether specific approaches generalize to a broad class of problems. • Striving to use procedures appropriately and efficiently. (NCTM, 2014, p. 47 -48)

Lens for Watching the Video As you watch the video again, attend to the

Lens for Watching the Video As you watch the video again, attend to the teacher actions that Build procedural fluency from conceptual understanding. Be prepared to give examples and to cite line numbers from the transcript to support your observations.

Watch the Video “Multiplication Strings”

Watch the Video “Multiplication Strings”

Video Observations In what ways did the teacher actions support students in building procedural

Video Observations In what ways did the teacher actions support students in building procedural fluency from conceptual understanding?

Effective Mathematics Teaching Practice Implement Tasks that Promote Reasoning and Problem Solving

Effective Mathematics Teaching Practice Implement Tasks that Promote Reasoning and Problem Solving

Implement Tasks that Promote Reasoning and Problem Solving Effective teaching of mathematics • provides

Implement Tasks that Promote Reasoning and Problem Solving Effective teaching of mathematics • provides opportunities for students to engage in solving and discussing tasks; • uses tasks that promote inquiry and exploration and are meaningfully connected to concepts; • uses tasks that allow for multiple entry points; and • encourages use of varied solution strategies. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 17)

Mathematical tasks can range from a set of routine exercises to a complex and

Mathematical tasks can range from a set of routine exercises to a complex and challenging problem that focuses students’ attention on a particular mathematical ideas (p. 17). It is important to note that not all tasks that promote reasoning and problem solving have to be set in a context or need to consume an entire class period or multiple days. What is critical is that a task provide students with the opportunity to engage actively in reasoning, sense making, and problem solving. . . (p. 20). National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

1. Work in pairs and select a string to study. 2. Identify a mathematics

1. Work in pairs and select a string to study. 2. Identify a mathematics learning goal for the string. 3. Anticipate ways students might reason as each equation is revealed. 4. Sketch area models you would use to represent the reasoning of students. 5. If time allows, study another string. Set A 2 x 8 5 x 8 7 x 8 String C 3 x 100 3 x 50 3 x 149 Set B 5 x 4 10 x 4 9 x 4 String D 200 x 5 200 x 20 200 x 25 198 x 25

Multiplication Strings In what ways do tasks such as multiplication strings support building procedural

Multiplication Strings In what ways do tasks such as multiplication strings support building procedural fluency from conceptual understanding while also engaging students in reasoning and problem solving?

Reflecting on the Effective Mathematics Teaching Practices

Reflecting on the Effective Mathematics Teaching Practices

As you reflect on the effective mathematics teaching practices examined in this session, summarize

As you reflect on the effective mathematics teaching practices examined in this session, summarize one or two ideas or insights you might apply to your own classroom instruction.